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COMPLIMENTS 

AMERICAN  BOOK  CO, 

A.  P.  OUNN,  Oen'l  Ag't, 

204  PINE  STREET, 

SAN   FRANCISCO. 


Digitized  by  the  Internet  Archive 

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•  •  ••  • 

•  •  •  • 


,   «•  •       •    •• 
.      •  •       •    ••• 


AMERICAN 


COMPREHENSIVE    ARITHMETIC 


BY 


M.    A.    BAILEY,   A.M. 

PBontssoR  or  Matiikmatics  in  tub  Kansas  Stats 
Normal  School,  at  Emporia,  Kansas 


NEW  YORK  :•  CINCINNATI  :•  CHICAGO 

AMKRICAN    BOOK    COMPANY 


•••••••  I  ■       / 

,.. -..  '  ^ 


OOPTSIOBT,  18W,  BT 
AMERICAN   BOOK  COMPANY. 

BA1UIY*8  kM..  OOMPE.   ABITH. 

EDUCATloVl  neP^» 


PREFACE 


In  this  work,  the  divisions  of  arithmetic  are  presented  in  their 
natural  order.  The  fundamental  operations  upon  integers,  common 
fractions,  decimals,  denominate  numbers,  and  numbers  expressed 
by  letters,  are  followed  by  their  applications  to  business  and  to 
various  employments.  The  introduction  of  the  chapter  on  literal 
quantities  is  somewhat  of  an  innovation,  but  is  in  accord  with 
the  views  of  leading  teachers,  and  is  based  upon  sound  principles. 
The  elementary  parts  of  algebra  should  be  studied  before  the 
advanced  parts  of  arithmetic,  because  they  are  more  easily  com- 
prehended, and  because  they  afford  valuable  assistanct  in  difficult 
problems.  For  the  same  reasons,  the  solution  of  many  classes  of 
problems  should  be  taught  by  algebraic  processes,  before  they  are 
taught  by  the  intricate  methods  required  by  analysis.  The  fact, 
too,  that  most  pupils  must  leave  school  quite  early,  is  a  strong 
reason  for  introducing  into  arithmetic  the  elements  of  algebra 
and  the  conclusions  of  geometry. 

From  the  beginning,  the  pupil  is  taught  to  develop  observa- 
tion and  thought  power.  Instead  of  being  required  to  memorize 
a  large  number  of  rules  and  directions,  he  is  encouraged  to  fix 
well  in  mind  the  result  to  be  obtained,  to  consider  carefully  the 

54f;»34 


PRKFACE 


means  at  his  command,  and  to  employ  those  means  according  to 
his  best  judgment. 

This  book  is  intended  to  complete  the  course  in  arithmetic 
required  by  district  and  city  schools. 


M.  A.  BAILEY. 


CONTENTS 


Integers  rA«v 

Notation  and  Numeration 7 

Addition 16 

Subtraction 28 

Multiplication 35 

Division 44 

Operations  Combined 60 

Factoring 68 

Common  Fractions 

Notation  and  Numeration 83 

The  Operations 84 

Analysis 98 

Decimal  Fractions 

Notation  and  Numeration 103 

The  Operations 107 

Special  Cases 114 

Denominate  Numbers  —  Enolish  System 

Notation  and  Numeration 119 

The  Operations 130 

Dknominate  Numbers  —  Metric  System 

Notation  and  Numeration 186 

The  Operations 142 

Literal  Quantities 

Notation  and  Numeration 145 

Addition 148 

Subtraction 150 

Multiplication 152 

Division 154 

Factoring 150 

Fractions 157 

Simple  Equations 159 

Two  Unknown  Quantities 167 

6 


6  CONTENTS 

PAOB 

pROPORTioir 170 

Solution  op  Problems 180 

Tercentage 

Simple  Cases 101 

Profit  ami  Loss 106 

Commission 108 

Taxes 200 

Trade  Discount 201 

Insurance 202 

Stoclts 206 

Interest 

Simple 217 

Annual 233 

Compound 234 

Involution  and  Evolution 

Involution 238 

Evolution 239 

Mensuration 

No  Dimension 240 

One  Dimension 249 

Two  Dimensions 251 

Tliree  Dimensions 260 

Similarity 268 

Proofs 272 

Occupations 

With  the  Lumber  Dealer 279 

With  the  Carpet  Dealer 285 

With  the  Paper  Hanger 286 

With  the  Mason 287 

With  the  Farmer 289 

Miscellaneous 

Longitude  and  Time 292 

Problems  in  Physics 294 

Accounts 295 

Definitions  and  Indkx 301 

Answers 311 


AMERICAN  COMPREHENSIVE  AMTHMETIO: 


NOTATION   AND   NUMERATION 


TERMS 

A  number  answers   the  question,   How  Illustration 

many  ?  How  many  apples  ? 

Numeration  is  the   process  of   reading  6. 

numbers.  6,  a  number,  or  integer. 

Notation  is  the  process  of  writing  num-  Read,  Six. 

bers.  Written,  6. 

NUMERATION 
In  reading  an  integer,  there  are  three  steps : 

The  number  is  pointed  off  from  right  to  left  into  periods  of  three 
figures  eoA^h. 

The  periods  are  named  from  right  to  left  to  learn  the  name  of  the 
left-hand  period. 

The  periods  are  read  and  named,  but  the  period,  000,  is  not 
read  and  units'  period  is  not  named. 

Pointing  off  into  periods  of  three  figures  each 
Beginning  aZ  the  right,  point  off  into  periods : 

1.  3600028.371.  4.       378002000028706954. 

2.  23600028371.  5.     4070080009000056207. 

3.  423600028371.  6.  53008700960057008301. 
7.   How  many  figures  may  there  be  in  the  left-liand  period  ? 

a   How  many   figures   must   there  be  in  each   of  the  other 
periods  ? 

Ex.  1.  3,600,028,871.      Ex.  2.  2.3,000,028,371.      Ex-  3.   423,600,028,371. 

7 


8  NOTATION   AND  NUMERATION 

m    '-i.   o      ' 

'      ^V*"'  Or        « 

Naming  the  periods 

tt.  1,023,438,019,000,320,078,736. 
6.  21,023,438,019,000,320,678,735. 
c.    321,023,438,019,000,320,678,735. 

from  right  to  left,  the  periods  are :  unitSy  thousands,  millions^ 
billions,  trillions,  quadrillions,  quintillions,  sextillions  .  .  .  . ;  from 
left  to  right:  ....  sextillions,  quintillions,  quadrillions,  trillions, 
billions,  millions,  thousands,  units. 

Higher  periods  after  sextillions  are :  septillionSj  octillions,  nonillions, 
decillions,  undecillions,  duodecillions,  .... 

9.  Memorize  the  names  of  the  periods  from  units  to  sextil- 
lions ;  from  sextillions  to  units. 

10.  In  a,  beginning  at  the  right,  point  to  and  name  each  period ; 
then  beginning  at  the  left,  point  to  and  name  each  period. 

11.  In  the  same  way,  point  to  and  name  the  periods  in  b  and  c. 

12.  In  the  same  way,  point  to  and  name  the  periods  in  exam- 
ples 1  to  6. 

Ex.  10.  Units,  thousands,  millions,  billions,  trillions,  quadrillions,  quin- 
tillions, sextillions ;  sextillions,  quintillions,  quadrillions,  trillions,  billions, 
millions,  thousands,  units. 

Reading  the  periods 

Each  period  is  made  up  of  three  orders,  units,  tens,  and  hundreds. 
III.  368  equals  3  hundreds,  6  tens,  8  units. 

In  units'  order,  except  when  1  is  in  tens'  order,  1,  2,  3,  4,  5,  6,  7,  8,  9  are 
read  one,  two,  three,  four.  Jive,  six,  seven,  eight,  nine. 

In  tens'  order,  2,  3,  4,  6,  6,  7,  8,  9  are  read  twenty,  thirty,  forty,  fifty, 
sixty,  severity,  eighty,  ninety.  1  in  tens'  order  is  read  with  the  units'  fic^ure. 
Thus:  10,  11,  12,  13,  14,  15,  16,  17,  18,  19;  ten,  eleven,  twelve,  thirteen,  four- 
teen, fifteen,  sixteen,  seventeen,  eighteen,  nineteen. 

In  hundreds'  order,  1,  2,  3,  4,  5,  6,  7,  8,  9  are  read  one  hundred,  two  hun- 
dred, three  hundred,  four  hundred,  five  hundred,  six  hundred,  seven  hundred, 
eight  hundred,  nine  hundred. 

0  in  any  order  is  never  read. 


INTEGERS  9 

Reading  the  periods 

d.  1,703,100,306,016,429. 

e.  25,603,007,075,018,208. 
/   436,000,056,000,001,200. 

13.  In  d,  read  units'  period ;  thousands'  period ;  millions'  period. 

14.  In  cf,  read  billions'  period ;  trillions'  period ;  quadrillions'. 

15.  In  e,  read  quadrillions'  period;  trillions'  period;  billions* 
period;  etc. 

16.  In  /,  read  each  period  in  succession,  beginning  at  the  left. 

Ex.  13.  Units'  period,  429,  four  hundred  twenty-nine.  4  in  hundreds' 
order  is  read  four  hundred;  2,  in  tens'  order,  twenty;  9,  m  units'  order, 
nine. 

Thousands'  period,  016,  sixteen.  0  is  not  read ;  1  in  tens'  order  is  read 
with  6,  the  figure  in  units'  order,  sixteen. 

Millions'  period,  306,  three  hundred  six.  3  in  hundreds'  order  is  read 
three  hundred;  0  is  not  read  ;  6,  in  units'  order,  six. 

The  process  as  a  whole 

17.  Read  37000016328. 

Ans.  37  billion,  16  thousand,  328.  Pointing  off  into  periods  of  three  fig- 
ures each,  37,000,016,328 ;  numerating  to  learn  the  name  of  the  highest 
period,  units,  thousands,  millions,  billions  ;  reading  and  naming  the  periods, 
87  billion,  16  thousand,  328. 

NoTR.  — The  plural  form  of  the  period's  name  should  be  used  in  numerating, 
bat  the  singular  form,  in  reading.    The  word  "  and  "  should  not  be  used. 

Read: 

20305070;  3005007001. 

10000001;  2000000002. 

88000050;  9120000878. 

6180700623074078. 

58097060078039007. 

136840050000200003. 

24.  3000045;  3000000452.       31.  2780010204009086981. 


la 

3678;  33678;  235678. 

25. 

19. 

2000;  20000;  200000. 

26. 

20. 

8008;  80008;  800008. 

27. 

21. 

7017;  70017;  70(K)17. 

2a 

22. 

6010;  70107;  701070. 

29. 

23. 

3124567;  1234567890. 

30. 

10  NOTATION    AND    NUMERATION 

NOTATION  —ARABIC 
In  writing  an  integer  there  are  two  steps : 

The  left-hand  pe)iod  is  tvritten  with  one,  two,  or  three  figures. 

The  other  periods  are  tJien  written  in  succession  with  three  fig- 
ures each. 

Writing  the  left-hand  period 

The  Arabic  notation  employs  two  devices : 

The  number  in  each  order  is  represent- 
ed by  one  of  the  symbols:   0,  1,  2,  3,  4,  6,      Three  hundred  eight, 
6,  7,  8,  9.  Z  hundred  0  ten  S, 

The  name  of  the  order  is  represented  by  ong 

relative  position. 

Write  as  a  left-hand  period : 

32.   Seven ;  seventy ;  seven  hundred ;  nine  hundred  sixty-eight. 

3a   Twenty;  sixteen;  thirty-six;  forty-five;  twelve;  five. 

34.  Ninety;  four  hundred;  five  hundred  sixty;  one  hundred 
one ;  nineteen ;  seventy-five. 

35.  Nine  hundred  ninety-nine;   two;    sixteen;   eighty;   forty- 
five;  thirteen. 

Ex.  32.    7  ;  70  ;  700  ; 

Writing  the  other  periods 

The  same  devices  are  employed  as  before,  but  each  period  must 
contain  three  figures.     See  examples  7  and  8. 

Write  as  a  full  period  : 

36.  Zero;  eight;  eighteen;  ninety;  fifty-six;  forty. 

37.  Thirty ;  one ;  six ;  seventy-five ;  one  hundred ;  three. 

3a  Five   hundred   six  ;    nine ;    seventeen  ;    sixty-eight ;   four ; 
seven;  seventy;  twenty. 

39.   Sixteen;  thirty-six;  forty-five;  five;  fourteen;  ninety-nine. 
Ex.36.    000;  008;  018; 


INTEGERS  11 

The  process  as  a  whole 

40.  Write  thirty-six  trillion,  one  hundred  seventy-six  million, 

six. 

Am.  36,000,176,000,006.  We  think  36  trillion,  and  write  36, ;  we  think 
no  billion,  and  write  000,  (the  work  now  appears  36,000,)  ;  we  think  one 
hundred  seventy-six  million,  and  write  176,  (36,000,176,)  ;  we  think  no 
thousand,  and  write  000,  (36,000,176,000,)  ;  we  think  six,  and  write  006. 
(36,000,176,000,006.) 

Write: 

41.  Two  billion,  seventeen  thousand,  one  hundred  twenty-six. 

42.  307   quadrillion,  20   billion,  four  hundred   seventy-seven. 

43.  300  sextillion,  4  trillion,  30   million,  98  thousand,  sixty. 

44.  65  quadrillion,  700  billion,  99  million,  999  thousand,  999. 

45.  Nineteen  million,  75  thousand,  seven  hundred  twenty-four. 

46.  Five  million,  7  hundred  20  thousand,  6  hundred  thirty. 

47.  30  quintillion,  300  trillion,  475  million,  4  thousand,  16. 
4a  555  sextillion,  505  million,  five  hundred   thousand,  500. 

49.  826  quadrillion,  469  billion,  8  million,  95   thousand,   18. 

Numeration  table 

50.  Memorize  the  following  table  both  by  orders  and  by  periods. 

By  Periods 

1000  units  =  1  thousand. 
1000  thou.  =  1  million. 
1000  mil.    =  1  billion. 
1000  bil.     =  1  trillion. 
1000  tril.     =  1  quadrillion. 
1000  quad.  =  1  quintillion. 


Ill  III  III  III  III  III  III  Ui 

368,   496.  743.  978.  432.  389.  986,  725. 

MX-        qnin-       qoad-      tril-         bil-         mil-        thoa-         a- 
tillionii  tillioni,  rillloni,    lioni,      Uout      lionf,      landi,      nita. 


Bv  Orders 

10  units 

=  1  ten. 

10  tens 

=  1  hundred. 

10  bund. 

=  1  thousand. 

10  thou. 

=  1  ten  thousand. 

10  t.  thou. 

=  1  hundred  thousand. 

10  h.  thoa 

,  =  1  million. 

.    . 

.... 

12  NOTATION  AND  NUMERATION 

NUMBERS  NAMED  — TO  A  THOUSAND 

In  naming  numbers  to  a  thousand,  it  is  thought  best  to  consider 
individuals  as  collected  into  groups  of  terij  ten  of  these  groups  as 
collected  into  a  higher  group,  and  ten  of  these  higher  groups  as 
collected  into  a  still  higher  group.  The  individuals  are  called 
wmYs;  the  groups,  orders.     Thus: 

Ten  units  are  considered  a  group  of  ten;  ten  teiw,  a  group  of  a  hundred; 
ten  hundreds,  a  group  of  a  thousand. 

The  individuals  forming  a  group  of  ten  are  given  independent 
names,  and  these  names  are  repeated  with  the  names  of  the 
orders  to  form  larger  numbers.     Thus : 

0?ip,  two,  three,  four.  Jive,  six,  seven,  eight,  nine, 
one  ten,  one  ten  one,  one  ten  two,  ....  one  ten  eight,  one  ten  nine, 
two  ten,  two  ten  one,   two  ten  two,  ....  two  ten  eight,  two  ten  nine, 

nine  ten,  nine  ten  one,  nine  ten  two,  ....  nine  ten  eight,  nine  ten  nine, 
one   hundred,    one    hundred    one,  ....  one  hundred  nine  ten  nine, 

nine  hundred,    nine  hundred  one,  ....  nine  hundred  nine  ten  nine. 

Note.  —  One  ten,  one  ten  one,  one  ten  two,  ....  two  ten,  two  ten  one 

have  been  abbreviated  to  ten,  eleven,  twelve twenty,     twenty-one,  .... 

From  the  above,  it  will  be  seen  that  numbers  to  a  thousand 
may  be  expressed  by  only  ten  different  symbols,  if  the  number  in 
each  order  is  expressed  by  a  figure,  and  if  the  names  of  orders 
are  expressed  by  relative  position.     Thus : 

90,  91,  92,  93,  94,  96,  96,  97,  98,  99, 
100,   101,    102,   103,    104,   105,    106,    107,    108, 197,    198,   199, 

900,   901,   902,   903,   904,   905,   906,   907,    908, 997,  998,  999. 

The  Arabic  notation  affords  these  devices,  and  therefore  har- 
monizes with  the  plan  of  naming  numbers.  The  oversight  of 
these  devices  by  the  Romans  and  most  other  nations  of  antiquity, 


nnrtr\-n 


.  +  o    fr^r.    +-k. 


inte(;krs  13 

numbers  named  — above  a  thousand 

When  the  number  of  individuals  is  larger  than  a  thousand, 
it  is  thought  best  to  consider  the  individuals  as  collected  into 
groups  of  a  thousand,  a  thousand  of  these  groups  as  collected 
into  a  higher  group,  a  thousand  of  these  higher  gi'oups  as  col- 
lected into  a  still  higher  group,  and  so  on.  The  individuals  are 
caMed  units ;  the  groups,  pmocfe.     Thus: 

A  thousand  units  are  considered  a  group  of  a  thousand ;  a  thousand  thou- 
sandSf  a  group  of  a  million;  a  thousand  millions,  a  group  of  a  billion;  and 
so  on. 

The  individuals  forming  a  group  of  a  thousand  are  named  as 
on  the  preceding  page,  and  these  names  are  repeated  with  the 
names  of  the  periods  to  form  larger  numbers.     Thus : 

1  thousand,  1  thou.  1, 1   thou.  999, 999  thou.  999, 

1  million,  1  mil.  1,  ....  1  mil.  999  thou.  999,  ....  999  mil.  999  thou.  999, 

From  the  above,  it  will  be  seen  that  numbers  larger  than  a 
thousand  may  be  expressed  by  only  ten  different  symbols,  if  the 
number  in  each  period  except  the  left-hand  is  written  with  three 
figures,  and  if  the  names  of  periods  are  expressed  by  relative 
position.     Thus : 

1,000;       1,001;      1,002;      1,003; 1,999; 999,999; 

1,000,000  ;   1,000,001 ;    1,000,002  ;  .  .  .  .  1,999,999  ; 9l)JM>99,999  ; 

Note.  — The  people  of  the  United  States  and  of  France  follow  the  foregoing 
plan  indefinitely ;  the  English,  only  to  a  million. 

When  the  number  of  individuals  is  larger  than  a  million,  the 
English  and  most  European  nations  think  it  best  to  consider  the 
individuals  as  collected  into  groups  of  a  milUony  a  million  of 
these  groups  as  collected  into  a  higher  group,  a  million  of  these 
higher  groups  as  collected  into  a  still  higher  group,  and  so  on. 
The  individuals  are  called  units;  the  groups,  periods.     Thus : 

A  million  units  are  considered  a  group  of  a  million ;  a  million  millions,  a 
group  of  a  billion  ;  a  million  billions,  a  group  of  a  trillion  ;  and  so  on. 


14  NOTATION   AND   NUMERATION 

NOTATION  — UNITED   STATES   MONEY 

Since  10  mills  make  1  cent,  10  cents  make  1  dime,  and  10 
dimes  make  1  dollar,  the  devices  already  explained  may  be 
employed  in  writing  United  States  money  if  it  is  agreed  that  the 
orders  from  the  left  shall  be  dollars,  dimes,  cents,  mills.    Thus : 

U.  S.  Five  dollars  no  dimes  eight  cents  three  mills 

U.  S.  5  dollars  0  dimes  8  cents  3  mills 

U.  S.  6083 

This  plan  is  modified,  however,  by  writing  the  symbol,  $, 
before,  and  a  period,  after,  the  number  of  dollars.     Thus : 

15.083. 

Write: 

51.  8  dollars  19  cents  2  mills.  56.  600  dollars  four  cents. 

52.  16  dollars  16  cents  4  mills.  57.  151  dollars  ninety  cents. 

53.  18  dollars  4  cents  6  mills.  5a  225  dollars  eight  mills. 

54.  425  dollars  sixty-two  cents.  59.  15  dollars  15  cents  5  mills. 

55.  521  dollars  38  cents.  60.  27  dollars  eleven  cents. 

Ex.  51.  $8,192.  19  cents  equals  1  dime  9  cents.  The  word  "dime"  is 
rarely  used  in  writing  or  reading. 

Bead  : 

61.  ^1.846;  $14.18;  $95.95.  66.  $43,065;  $18.03;  $15.92. 

62.  $91.12;  $84.98;  $41.05.  67.  $91,164;  $37.33;  $41.25. 

63.  $82.07;  $46.04;  $37.07.  68.  $126,003;  $5.05;  $23.50. 

64.  $86.06;  $39.09;  $27.64.  69.  $8971.04;  $3.07;  $94.01. 

65.  $51.22;  $49.87;  $91.14.  70.  $6128.97;  $9.21;  $79.49. 

Ex.  61.  1  dollar  84  cents  6  mills.  The  natural  reading  would  be,  1  dollar 
8  dimes  4  cents  6  mills,  but  it  is  customary  to  read  dimes  and  cents  together 
as  cents. 


INTEGERS 


16 


NOTATION  — ROMAN 

The  Roman  Notation  uses  seven  capital  letters,  viz. :  I,  1 ;  V, 
6;  X,  10;  L,  50;  C,  100;  D,  500;  M,  1000. 

Since  it  does  not  use  an  independent  symbol  for  each  of  the 
first  ten  numbers,  it  is  out  of  harmony  with  the  plan  of  naming 
them,  and  is  rai-ely  used  except  for  ornamental  purposes. 

Independent  names  are  given  to  one,  two,  three,  four,  five,  six,  seven, 
eight,  nine,  ten;  the  Roman  notation  uses  independent  symbols  for  one^ 
Jive,  ten,  fifty,  one  hundred,  five  hundred,  and  one  thoitsand. 


The  Roman  notation  employs  these  devices  : 


Repeating  a  letter  repeats  its  value. 

Wheti  a  letter  is  placed  after  one  of 
greater  value,  its  value  is  to  be  added  to 
that  of  the  greater. 

When  a  letter  is  placed  before  one 
of  greater  value,  its  value  is  to  be  sub- 
tracted from  that  of  the  greater. 

When  a  letter  is  placed  between  two 
letters,  each  of  greater  value,  its  value  is  to 
be  subtracted  from  the  sum  of  the  other 
two. 

A  bar  placed  over  a  letter  multiplies 
its  value  by  1000. 


Ill,  3  ;  CC,  200 ;  XXX,  30. 


XI,  11 ;  LXXX, 


IX,  9  ;  CD,  400. 


XIV,  14  ;  DXL,  640. 


X,  10,000 ;  CMIII,  900,008. 


Express  by  the  Arabic : 

71.  IT,  XX,  CCC,  MMM. 

72.  LI,  CV,  DC,  LXX. 
7a  IV,  XL,  XC,  CD,  CM. 
74.  XIX,  CXL,  DXC,  MCD. 
75   D,  M,  CDCCCXXXIX. 


Express  by  the  Roman : 

76.  1,2,3,4,5,6,7. 

77.  8,  9,  10,  11, 12, 13. 
7a  28,  83,  95,  101, 190. 
79.  256,379,400,568. 

aa  1492,  1520,  1876,  1896. 


ADDITION 


TERMS 

The  whole  is  equal  to  the  sum  of  all  its  parts.     Thus : 

9  cents  =  4  cents  -f  5  cents. 
If  the  whole  is  wanting,  this  becomes 

what  =  4  cents  +  5  cents  ? 
It  means  "  what  is  the  sum  of  4  cents  and  5  cents  ?  " 

Addition  is  the  process  of   uniting  two  or  4) 

more  numbers  into  one.     The  numbers  to  be  f  I       ^'"^ 

united  are  addends;   the  result,  the   sum,  or  5,  sum. 

amount.  6  +  4  =  10, 

The  sign  of  addition  is  +.  rea^^ 

The  sign  of  equality  is  =.  6  plus  4  equals  10. 


COUNTING  BY  ONES 

Addition  may  be  performed  by  counting  by  ones. 

1.  In  this  way,  find  the  sum  of  4,  3,  and  2. 

Counting  4,  then  3,  then  2,  and  making  a  mark  at  each  count,  we  have 
////,  ///,  //;  counting  these  together,  we  have  one,  two,  three,  four,  five,  six, 
seven,  eight,  nine. 

2.  Counting  by  ones,  find  the  sum  of  5  and  6. 

3.  Counting  by  ones,  find  the  sum  of  8,  9,  and  7. 

4.  Would  you  care  to  find  the  sum  of  968  and  754  in  this  way  ? 
Why  not? 

16 


INTEGERS  17 

COMMON  METHOD 

There  are  forty-five  combinations  of  the  first  nine  numbers 
taken  two  and  two.     They  should  be  memorized  as  wholes. 

5.  Memorize  the  follomng  : 

1         2         23  34  345  466 

1,  ;?;   1,  3;   2,  1,  4;   2,  1,  5;   3,  2,  1,  6;     3,  2,  1,  7 

4667       6678       66789 
4,  3,  2,  1,  8;  4,  3,  2,  1,  P;    6,  4,  3,  2,  1,  10 

6789       6789  789 

6,  4,  3,  2,  ii;    6,  6,  4,  3,  7j?;       6,  6,  4,  i5 

78  9      89      89       9      9 

7,  6,  6,  i^;   7,  6,  i5;   8,  7,  75;   8,  i7;   9,  18, 

Thus :  4,  8,  2t  i«  are  different  ways  of  expressing  8. 

Call  the  sums  : 

83927948697179696 

6.  6,  2,  3,  2,  8,  7,  4,  1,  4,  2,  7,  1,  6,  1,  6,  4,  4. 

97868196939482178 

7.  8,  9,  7,  9,  8,  4,  9,  6,  2,  9,  6,  9,  3,  8,  6,  4,  9. 

63828346567964762 
a  6,  3,  4,  6,  3,  1,  2,  2,  3,  3,  6,  6,  1,  3,  2,  2,  4. 

19621674896271799 
9.  6,  8,  6,  6,  7,  8,  1,  1,  6,  4,  7,  1,  6,  2,  7,  6,  3. 

82938986761459769 
la  7,  9,  3,  7,  6,  9,  2,  1,  3,  9,  8,  7,  1,  6,  6,  8,  8. 

29836948693172698 

11.  3,  2,  4,  6,  6,  7,  6,  1,  6,  3,  6,  3,  6,  7,  6,  4,  4. 

67489617879483979 

12.  7,   1,   6,   6,   4,   6,    9,   6,   8,    7,   6,   8,    1,   8,    7,   8,   2. 

Ex.  6.    18,  6,  12,  4,  16. 

AMBR.    ARirn.  —  2 


18  ADDITION 

Steps  in  adding 
Is  the  sum  of  the  units  more  tJuxn  9 : 

13.  When    6   is  added  to    24,   23,   27,   25,   28,   36,   41,   32? 

14.  When    5   is  added  to    12,   35,   38,   27,   43,   54,    19,    21? 
Ex.  13.   Yes,  no,  yes,  yes,  .... 

Declare  the  tens  of  the  sum  : 

15.  When    36    is  increased  by    9,    2,    8,    4,    6,    3,    1,    5,    7. 

16.  When    24    is  increased  by   8,    7,    2,    9,    1,    3,    6,    5,    4. 

17.  When  68  is  increased  by  5,  9,  2,  8,  4,  3,  7,  1,  6. 
la  When  79  is  increased  by  8,  3,  1,  4,  9,  5,  6,  7,  2. 
19.  When  27  is  increased  by  5,  9,  2,  8,  4,  3,  6,  1,  7. 
2a  When  35  is  increased  by  3,  1,  7,  2,  6,  4,  9,  6,  8. 

Ex.  15.   Forty,  thirty,  forty,  forty, .... 

Declare  the  units  of  the  sum  : 

21.  When   9   is  added  to   15,    29,  37,  46,  53,  22,  41,  68. 

22.  When  8  is  added  to  94,  38,  75,  26,  52,  63,  89,  71. 
2a  When    7   is  added  to   35,    48,  84,  76,  92,  53,  69,  21. 

24.  When  6  is  added  to  20y  38,  47,  56,  63,  72,  51,  49. 

25.  When  5  is  added  to  17,  23,  31,  45,  56,  62,  34,  49. 
2e  When  4  is  added  to  14,  58,  75,  66,  82,  91,  27,  19. 
27.  When  3  is  added  to  18,  16,  27,  39,  45,  64,  23,  12. 

Ex.  21.    4,  8,  6,  5,  2, 

2a  Declare  the  sums  in  examples  13  to  27  inclusive. 
Ex.  13.   30,  29,  33,  31, 

29.   Count  from  1  to  100  by  9's ;  by  8's ;  by  7's ;  by  6's ;  by  5's ; 
by  4's ;  by  3's ;  by  2's. 

Ex.  29.   By  9's ;  1,  10,  19,  28,  37, 


INTEGERS  19 

A  single  order 

51.    52.   53.   54.    55.   56.    57.  58.    59.  60.  61.  62.  63.  64.  65. 

30.  8,     (5,     9,     3,     4,     7,     8,  4,     9,  5,  3,  8,  7,  6,  8. 

31.  6,    4,    8,    2,    6,    6,    4,  6,     9,  8,  8,  6,  4,  9,  8. 

32.  6,    6,    9,    9,    8,    4,    7,  6,    8,  8,  7,  9,  4,  8,  3. 

33.  6,    7,    8,    9,    4,    6,    8,  9,    8,  7,  6,  5,  4,  3,  2. 

34.  4,    4,    5,    5,    6,    6,     7,  7,    8,  8,  9,  9,  8,  8,  7. 

35.  3,    6,    9,    6,    3,    2,    4,  8,    4,  2,  5,  7,  9,  8,  6. 

36.  6,     7,     9,     8,     8,     4,     7,  6,     9,  8,  3,  4,  7,  7,  9. 

37.  9,    4,    3,    2,    8,     7,    6,  3,    9,  4,  8,  6,  3,  3,  6. 
3a    6,    8,    3,    9,    7,    4,    2,  5,    8,  7,  7,  4,  7,  6,  8. 

39.  8,     8,     4,     4,     6,     6,     5,  8,     9,  7,  6,  3,  7,  4,  8. 

40.  8,     9,    7,     6,    4,    3,    9,  8,    2,  6,  3,  8,  9,  4,  3. 

41.  6,    6,    4,    3,    3,    4,    8,  7,    6,  9,  4,  6,  8,  7,  6. 

42.  6,    6,    6,    3,    7,    2,    8,  1,    9,  4,  7,  9,  3,  8,  7. 

43.  2,    2,    8,    6,    5,    4,     7,  9,    8,  6,  3,  6,  5,  7,  4. 

44.  8,    8,    6,    5,    3,    2,    9,  4,    7,  6,  3,  8,  7,  6,  3. 

45.  8,    6,    5,    3,    9,    8,    6,  6,    4,  8,  8,  5,  7,  2,  3. 

46u    2,    9,     1,    3,    8,    2,    5,  3,    7,  6,  4,  7,  5,  5,  6. 

47.    2,    3,    4,    6,    6,     7,    8,  9,     8,  7,  6,  5,  4,  3,  2. 

4a    4,    6,    6,    4,    7,    8,    6,  7,    6,  3,  2,  9,  7,  6,  6. 

49L    1,     1,     3,     5,     6,     2,     3,  4,     7,  1,  1,  8,  2,  5,  4. 

50.    7,    6,    3,    2,    9,    8,    8,  4,    3,  8,  7,  6,  3,  1,  9. 

Ex.  30.   8,14,  23,  26,  30,  37,  ...  .  Do  not  say  ''8  and  6  are  14,  and  U 
are  23." 

Ex.  31.   By  this  method,  we  add  one  figure  at  a  time,  but  when  the  sum 
of  two  adjacent  addends  is  10,  it  is  customary  to  add  tl»e  two  at  once ;  we 

say  10,  20,  26,  35,  41, 


20  ADDITION 

More  than  one  order 
66.  Add  and  explain  :  368,  496,  709. 


368  The  sura  of  the  units  is  23  units,  or  2  tens 

iQQ       ts  and  3  units ;  we  write  3  in  units'  column  and 

_,  Q       17  carry  2  to  tens'  coluran. 

^*  The  sum  of  the  tens  is  17  tens,  or  1  hundred 

1573  *"^  ^  tens;  we  write  7  in  tens'  column  and 

carry  1  to  hundreds'  column  ;  etc. 


To  provcy  add  in  the  opposite  direction.     See  p.  79. 

Note. —For  convenience  in  proving,  the  sum  of  each  column  should  be  written 
by  itself  as  soun  as  it  is  obtained. 

67.  Add  and  explain :  468,  963,  735,  843,  987,  638,  475,  327. 
6a  Add  and  explain :  745,  908,  722,  496,  783,  954,  738,  873. 


69. 

70. 

71. 

72. 

73. 

74. 

777 

398 

666 

999 

995 

672 

787 

846 

694 

888 

881 

547 

898 

946 

784 

765 

654 

857 

798 

775 

996 

877 

343 

351 

879 

238 

886 

895 

637 

893 

878 

876 

259 

788 

448 

752 

788 

439 

448 

399 

549 

695 

988 

777 

679 

757 

870 

452 

899 

989 

876 

499 

626 

537 

789 

646 

947 

787 

765 

794 

787 

988 

589 

696 

693 

686 

877 

568 

487 

789 

247 

964 

889 

859 

444 

789 

692 

415 

787 

363 

668 

947 

631 

213 

978 

496 

843 

787 

786 

429 

475 

132 

246 

987 

325 

737 

13i^ 

547 

745 

579 

417 

217 

824 

724 

993 

813 

629 

425 

INTEGERS 


21 


COUNTING  BY  COMBINATIONS   10-19 

The  combinations  of  three  or  more  digits  whose  sums  are  10  to 
19  should  be  learned  so  that  the  results  can  be  recognized  and 
called  at  a  glance. 


8 
S 

Thus:  8 

should  be  recognized 

5 

ass, 

4 
2 

i5;  8  as  9,  15; 

etc. 

Declare  the  sums  rapidly  : 

8 

1 

8 

8 

8 

7 

6 

6 

4 

4 

1 

9 

7 

8 

7 

3 

3 

4 

5 

4 

7 

1 

75.   2 

_9 

2 

2 

3 

1 

3 

9 

6 

8 

8 

7 

8 

4 

6 

7 

8 

6 

4 

9 

7 

9 

3 

5 

4 

3 

4 

2 

3 

6 

8 

3 

5 

76.    6 

3 

3 

2 

1 

2 

5 

9 

1 

4 

4 

6 

6 

8 

9 

7 

9 

6 

7 

7 

6 

9 

3 

5 

4 

3 

3 

1 

3 

4 

5 

3 

3 

77.    2 

4 

1 

1 

2 

2 

3 

4 

4 

1 

2 

4 

5 

5 

6 

6 

7 

7 

4 

6 

4 

6 

6 

4 

5 

2 

1 

3 

2 

6 

1 

4 

3 

6 

2 

4 

5 

3 

7 

3 

2 

2 

1 

5 

7a   3 

2 

5 

1 

4 

2 

2 

7 

2 

2 

2 

5 

6 

6 

6 

4 

2 

5 

4 

1 

2 

3 

2 

1 

3 

1 

4 

7 

3 

1 

8 

4 

6 

1 

1 

4 

1 

1 

1 

1 

1 

1 

1 

4 

79.   3 

2 

2 

4 

2 

1 

3 

4 

6 

3 

4 

7 

6 

6 

9 

3 

1 

3 

4 

1 

4 

2 

1 

3 

1 

1 

3 

8 

4 

6 

8 

4 

6 

1 

1 

1 

4 

2 

2 

2 

8 

4 

2 

4 

80.    4 

2 

7 

5 

2 

7 

5 

1 

4 

4 

4 

Ex.  75. 

19,  17, 

18,  17, 

.  .  .  . 

Ex.  78. 

19. 13 

,19. 

13,  .  .  . 

. 

22  ADDITION 

Steps  in  adding 
Is  the  sum  of  the  units  more  than  9 : 

81.  When    15    is  added  to    72,  68,  47,  31,  45,  23,  36,  39? 

82.  When    18    is  added  to    94,  47,  39,  28,  55,   63,  92,  61? 
Ex.  81.  No,  yes,  yes,  no,  yes,  .... 

Declare  the  tens  of  the  sum : 

8a  When  46  is  increased  by  13,  16,  17,  12,  18,  14,  19. 

84.  When  38  is  increased  by  15,  19,  17,  12,  14,  16,  13. 

85.  When  92  is  increased  by  17,  19,  18,  14,  16,  12,  15. 
8a  When  29  is  increased  by  11,  15,  18,  l6,  19,  17,  12. 
87.  When  37  is  increased  by  17,  14,  16,  19,  13,  14,  11. 
8a  When  55  is  increased  by  12,  19,  17,  16,  13,  14,  18. 

Ex.  83.   Fifty,  sixty,  sixty,  fifty,  sixty,  sixty,  .... 

Declare  the  units  of  the  sum : 

89.  When  17  is  added  to  71,  85,  96,  84,  23,  72,  99,  87. 

90.  When  16  is  added  to  15,  29,  37,  46,  53,  22,  41,  74. 

91.  When  13  is  added  to  17,  19,  26,  28,  35,  41,  63,  94. 

92.  When  14  is  added  to  68,  94,  85,  96,  47,  53,  72,  91. 

93.  When  18  is  added  to  29,  37,  46,  55,  83,  92,  24,  38. 

94.  When  12  is  added  to  24,  69,  78,  37,  43,  82,  91,  65. 
Ex.  89.  8,  2,  3,  1, 

95.  Declare  the  sums  in  examples  81  to  94,  inclusive. 
Ex.  81.   87,  83,  62,  46, 

9a   Count  from  1  to  200  by  19's;  by  18's;  by  17's;  by  16's ; 
by  15's ;  by  14's  ;  by  13's  ;  by  12's ;  by  ll's. 
Ex.  96.    By  19's.     1,  20,  39,  58,  77,  96,  ...  . 


INTEGERS  23 

Addends  less  than  20 

121.  122.  123  124.  125.  126.  127.  12a  129.  13a  131. 

97.  13,  14,  15,  16,  17,  18,  12,  14,  11,  19,  18. 

9a  12,  16,  13,  18,  11,  17,  15,  16,  13,  18,  11. 

99.  15,  17,  12,  13,  18,  11,  16,  15,  12,  14,  17. 

100.  19,  13,  14,  17,  14,  19,  13,  17,  15,  12,  14. 

101.  16,  11,  15,  14,  17,  12,  14,  13,  17,  11,  13. 

102.  17,  14,  18,  19,  15,  13,  17,  19,  16,  15,  12. 

103.  18,  19,  17,  12,  13,  14,  19,  18,  14,  17,  19. 

104.  19,  16,  14,  13,  .  12,  18,  15,  16,  17,  19,  16. 

105.  11,  19,  15,  13,  12,  16,  17,  13,  19,  18,  16. 
loa  16,  18,  14,  12,  10,  11,  13,  15,  17,  19,  16. 
107.  19,  15,  13,  12,  11,  18,  16,  14,  13,  18,  17. 
lOa  16,  16,  15,  14,  19,  17,  16,  17,  18,  14,  13. 

109.  13,  16,  19,  19,  17,  17,  15,  15,  11,  11,  17. 

110.  18,  18,  16,  16,  13,  19,  17,  11,  13,  14,  14. 

111.  17,  17,  16,  14,  15,  13,  19,  17,  15,  11,  12. 

112.  15,  14,  12,  10,  11,  12,  13,  14,  17,  19,  18. 
ua  16,  12,  16,  16,  15,  14,  18,  19,  17,  16,  15. 
U4.  11,  13,  15,  17,  19,  17,  15,  13,  11,  12,  14. 
115.  16,  17,  19,  16,  11,  13,  16,  14,  13,  17,  16. 
ua  17,  17,  16,  16,  14,  12,  11,  19,  16,  15,  13. 
117.  14,  13,  19,  16,  17,  12,  16,  16,  13,  19,  18. 
lia  12,  11,  18,  16,  14,  13,  17,  19,  16,  16,  17. 
119.  19,  18,  17,  16,  15,  11,  16,  16,  12,  11,  16. 
12a  17,  16,  14,  13,  12,  16,  16,  11,  17,  19,  16. 

Ex.  97.   13,  27,  42,  68,  76,  93, 105, 


24 


ADDITION 


Addends  greater  than  19 

132.   By  combinations  10  to  19,  add:  24,  72,  98,  69,  34,  73,  28, 
43,  52,  89,  73,  58. 

8 
8  2 

We  see  s  as  11,  and  say  11  ;  we  see  9  as  14, 

4 

8  8 

and  say  25 ;  »  as  15,  and  say  40 ;  9  as  17,  and 


say  57  ;  2  as  6,  and  say  63. 


We  see  the  6  to  carry  and  5  as  18,  and  say  18 ; 

6 
4  8 

6  7 

we  see  8  as  17,  and  say  35 ;  2  as  18,  and  say  53  ; 

8 

9  as  18,  and  say  71. 

Note.  — The  explanation  may  be  given  as  on 
p.  20 ;  the  sum  of  the  units  is  63  units,  or  6  tens  and 
3  units ;  etc. 


138. 

938 
544 
629 
887 
368 
299 
886 
743 
592 
447 
918 
925 
759 
967 


721 
i98] 

(69 

S4] 

6S 

73 

^28] 

[4S]           71 

62 

[89] 

\7S^ 
68] 

,00 

713 

c^ 

@    ( 

625 

198 

419 

344 

"W  ■ 

137 

333 

582 

548 

991 

727 

377 

949 

^m 

194 

454 

4tr 

643 

798 

249 

386 

818 

666 

726 

977 

638 

293 

247 

586 

393 

269 
475 
628 
849 
137 
966 
294 
888 
376 
789 
649 
555 
228 
494 
347 
325 


^i3ei 

597 

487 
676 
345 
799 
288 
376 
467 
156 
165 
246 
254 
334 
343 
628 
627 


,^37. 

896 
288 
144 
727 
419 
589 
476 
665 
647 
925 
194 
886 
827 
889 
929 
588 


484 


INTEGERS 

25 

Practice  for  any  method 

140. 

141. 

142. 

143. 

144. 

63954 

611043 

545982 

662347 

845954 

87445 

626915 

606819 

257938 

358346 

]:;•_".>  I 

537454 

146608 

689477 

214324 

:>1G87 

442014 

889478 

893298 

843567 

91532 

780894 

395777 

478469 

413578 

72927 

122993 

865092 

328947 

495219 

51470 

726915 

323459 

886539 

254183 

82044 

484471 

478949 

445328 

145482 

94514 

705606 

338948 

768894 

562531 

12592 

560247 

667848 

325537 

214644 

88416 

827922 

769949 

892468 

523671 

97654 

439706 

876329 

729848 

954751 

92214 

364399 

448869 

849889 

552169 

82405 

671361 

732964 

946677 

773713 

96185 

225163 

985263 

833849 

631526 

:>5391 

954670 

868897 

486695 

652847 

69039 

258520 

402986 

329875 

156879 

71623 

184109 

399478 

684498 

235659 

89555 

194020 

456984 

725578 

921982 

66526 

184059 

737842 

398765 

567399 

68507 

680637 

698325 

438942 

936783 

21661 

457351 

447983 

668937 

356773 

92717 

721295 

892567 

865432 

692683 

•S329 

665260 

443846 

598428 

578756 

31786 

539891 

329847 

697646 

578325 

97394 

382579 

666744 

393948 

565731 

86929 

426869 

889847 

872896 

757785 

31993 

389495 

653894 

498695 

567320 

51395 

708040 

655;«3 

213151 

315625 

r:i:.::. 

943511 

773311 

431512 

567921 

1  Ic... 

443225 

773221 

772202 

521673 

26  ADDITION 

PROBLEMS 
First  fonn  of  analysis 

145.  If  there  are  96  apples  in  one  pile  and  88  apples  in  another, 
how  many  are  there  in  both  ? 

OR 

gg  In  both,  there  are  the  sum  of  96  apples  and 

88  apples,  or  184  apples. 

To  Moritt  vohai  each  term  represents  is  of  service. 

146.  A  man  paid  $  45  for  a  horse,  and  $  95  more  for  a  carriage 
than  for  the  horse.     Hfew  much  did  he  pay  for  both  ? 

$  M5i  horse 
95    increase  "^^^  carriage  cost  the  sum  of  $45  and  $95, 

r—r'  or  $  140  ;  both  cost  the  sum  of  $  45  and  $  140, 

140y  carriage  or  $186. 

185y  both 

i^  A  drawing  is  often  of  service. 

147.  One  village  is  37  miles  west  and  another  48  miles  east 


t\ 


of  Denvor.     Mow  far  apart  are  the  villages? 
4 


D  B 


Since  it  is  37  miles  from  A  to  D,  and  48 
^j  A  to  D  miles  from  D  to  B,  from  A  to  B  it  is  the  sum, 

4Sy-^  to  B  or  86  miles. 

85,  A  to  B 

148  A  T>inn  i^nid  350  for  a  horse,  $125  for  a  carriage,  $12  for 
a  hai  '  5  for  a  saddle.     How  much  did  he  pay  in  all  ? 

149.  The  village  A  is  36  miles  east  of  Chicago,  and  B  is  43 
miles  east  of  A.     How  far  is  it  from  Chicago  to  B  ? 

150.  A  grocer  sells  18  pounds  of  coffee,  16  pounds  more  of  tea 
than  of  (  d  96  pounds  more  of  sugar  than  of  tea.  How 
many  pou..-  ;1  (loes  he  sell  ? 

151.  A  has  1  *»  has  $  29  more  than  A ;  C  has  as  much  as 
A  and  B  ar  :  and  D  has  as  much  as  A,  B,  and  C 
together.     H'                     all  possess? 


INTEGERS  27 

152.  How  many  strokes  are  made  in  a  day  by  a  clock  which 
strikes  the  hours  ? 

153.  Mr.  Brown  owes  three  bills,  one  for  $1987,  another  for 
$  1849,  and  a  third  for  $  2789.     How  much  does  he  owe  in  all  ? 

154.  A  man  bought  4  horses,  paying  for  them  $  3985,  $  5025, 
$2789,  and  $6898;  he  sold  them  for  $2399  above  cost.  How 
much  did  he  receive  for  them  ? 

155.  A  man  paid  $  348G  for  a  lot,  $  2878  for  a  house,  $  1695 
for  furniture,  and  $387  for  repairs;  he  disposed  of  his  entire 
property  for  $285  above  cost.     How  much  did  he  receive? 

156.  A  nurseryman  sold  862  pear  trees,  965  more  apple  trees 
than  pear  trees,  688  peach  trees,  466  more  plum  trees  than  apple 
trees,  and  568  ornamental  trees.  Find  the  entire  number  of  trees 
sold. 

157.  The  village  A  is  six  hundred  sixty-five  miles  east  of 
Chicago,  and  B  is  eight  hundred  eighty  miles  west  of  Chicago. 
What  is  the  distance  from  A  to  B  ? 

15a  The  Duke  of  Wellington's  army  at  Waterloo  consisted 
of  26,661  English  infantry,  8735  English  cavalry,  6877  English 
artillery,  and  33,413  allies.     How  large  was  his  army  ? 

159.  North  America  has  an  area  of  9,349,741  square  miles ;  and 
South  America,  6,887,794.  What  is  the  area  of  the  American 
continent  ? 

160.  Mt.  Etna  is  10,874  feet  above  the  level  of  the  sea;  Mt. 
lUanc  is  4870  feet  higher  than  Mt.  Etna;  Mt.  Everest  is  13,258 
feet  higher  than  Mt.  Blanc.     What  is  the  height  of  Mt.  Everest  ? 

161.  A  train  travels  719  miles  one  day,  698  miles  a  day  for  the 
next  three  days,  692  miles  a  day  for  two  days,  and  718  miles  the 
seventh  day.     How  many  miles  do6  it  travel  during  the  week? 

162.A-  farmer  raised  on  one  fjitib  t<Mi  hundred  thirty-eight 
bushels  of  wheat  and  three  hundn'd  two  bushels  of  barley;  on  a 
second  farm  one  thousand  one  bushels  of  wheat  and  five  hundred 
two  bushels  oflp&t8.'4idBt  wA  his  entire  crop  of  grain? 


SUBTRACTION 


TERMS 


9  cents  =  4  cents  4-  5  cents. 
If  the  second  addend  is  wanting,  this  becomes 

9  cents  =  4  cents  +  what. 
This  is  commonly  written 

what  =  9  cents  —  4  cents  ? 
It  means  "  what  is  the  other  of  two  numbers  when  one  of  them 
is  4  cents  and  their  sum  9  cents,  or  what  is  the  difference  between 
9  cents  and  4  cents  ?  " 


4,  subtrahend. 

5,  remainder. 
9-4  =  5, 


Subtraction  is  the  process  of  finding  the        One  of  two  numbers 

other  of  two  numbers,  when  one  of  them     '^  ^'  ^''^  ^^^'^  8^°^'  ^• 

J    . ,    .  .  Find  the  other, 

and  their  sum  are  given. 

The    sura   is   the   minuend;    the  number  9,  minuend. 

given,  the  subtrahend;  the  number  required, 
the  difference,  or  the  remainder. 

The  sign  of  subtraction  is  — .  '    ^^^^ 

9  minus  4  =  6. 
COUNTING  BY  ONES 

Subtraction  may  be  performed  by  counting  by  ones. 

1.  In  this  way,  subtract  5  from  8. 

Counting  8  and  making  a  mark  at  each  count,  ////////;  counting  5  and 
crossing  a  mark  at  each  count,  XXXXX/Z/j  counting  what  is  left,  we  have  3. 

2.  Counting  by  ones,  find  the  difference  between  9  and  12. 

3.  Counting  by  ones,  find  the  remainder  when  11  is  subtracted 
from  15. 

4.  Would  you  care  to  subtract  986  from  2384  in  this  way? 
Why  not? 


INTEGERS  29 

COMMON   METHOD 

If  the  forty-five  combinations  in  addition  have  been  mastered, 
the  results  in  the  following  examples  may  be  called  rapidly.  In 
addition,  we  have  two  addends  to  name  the  sum ;  in  subtraction, 
the  sum  and  one  addend  to  name  the  other. 

5.   Given  the  sum  and  one  addend,  name  the  other: 
1  12  132  1324  2543 

36542  327546  4753286 

999997.      999999  J?.      99999990 

.,.,.,.,.,/,  .,.,.,.,.,.,0,  .,.,.,.,.,.,.,£/ 

729354861      26374958 
999999999  /n.  99999    '^99    If 

If     .,  .,  .,  .,  .,  .,  .,  .,  J.U  J  .,  .,  .,  .,  .,  .,  .,  .,  IJ. 

3849576     475968     987 
9999999  10.    9    9    9    9    9    9    i<i'    9    9    9     ifi 

58976      6897      98      9 

?,  ?,  ?,  ?,  ?,  i^;   ?,  ?,  ?,  ?,  /5;   ?,  ?,  77;   ?,  i<^. 

Thus:  1;  2,  1  ;  3,  1,  2;  ...  . 
Declare  the  remainders : 


18 

14 

12 

10 

12 

11 

17 

15 

16 

12 

6. 

9 

5 

3 

4 

5 

3 

9 

8 

8 

4 

11 

10 

17 

13 

16 

15 

10 

10 

11 

13 

7. 

8 

2 

8 

6 

9 

7 

9 

1 

7 

7 

14 

13 

16 

13 

11 

11 

15 

13 

11 

9 

a 

6 

8 

7 

5 

5 

9 

9 

4 

4 

6 

13 

14 

15 

14 

12 

11 

9 

9 

7 

4 

9. 

9 

8 

6 

9 

8 

2 

4 

3 

3 

3 

10 

14 

10 

11 

10 

12 

10 

12 

10 

8 

10. 

8 

7 

6 

6 

7 

9 

5 

6 

3 

1 

Ex.  6.  9,  9,  9,  6,  7,  ...  .     Do  not  say,  ♦♦  9  from  18  leaves  9." 


80  SUBTRACTION 

SUBTRAHEND   GREATER   THAN   NINE 

Method  A 

Adding  10  to  any  order  of  the  minuend  and  1  to  the  next 
higher  order  of  the  subtrahend,  cannot  affect  the  remainder. 

U.   From  501  subtract  398  and  explain. 

501  In  fill.    8  units  from  1  unit  we  cannot  take  ; 

ggg  we  add  10  units  to  1  unit  making  11  units,  and  1 

ten  to  9  tens  making  10  tens;  8  units  from  11 

lOS  units  leaves  3  units  ;  we  write  3  in  units*  column. 

10  tens  from  0  tens  we  cannot  take  ;  we  add  10 
tens  to  0  tens  making  10  tens,  and  1  hundred  to  3  hundreds  making  4  hun- 
dreds ;  10  tens  from  10  tens  leaves  0  tens  ;  we  write  zero  in  tens'  column  ;  etc. 

Abbreviated.  3,  0,  1.  We  look  upon  1  as  11  and  say  3  ;  we  look  upon 
9  as  10,  and  0  as  10,  and  say  0  ;  we  look  upon  3  as  4  and  say  1. 

Note.  —  It  is  a  great  loss  of  time  to  say,  "  8  from  11  leaves  3."  While  these 
words  are  being  said,  the  process  is  delayed. 

Method  B 

Taking  1  from  any  order  of  the  minuend  and  adding  its 
equivalent  to  any  other  order  of  the  minuend,  cannot  affect  the 
remainder. 

U.  From  501  subtract  398  and  explain. 

5Q1  In  full.     8  units  from  1  unit  we  cannot  take ; 

«Q«  we  take  1  hundred  from  5  hundreds  leaving  4  hun- 

dreds ;    1  hundred,  or  10  tens,  with  0  tens,  10 

X03  tens  ;  we  take  1  ten  from  10  tens  leaving  9  tens ; 

1  ten,  or  10  units,  with  1  unit,  11  units;  8  units 
from  11  units  leaves  3  units  ;  we  write  3  in  units'  column. 

9  tens  from  9  tens  leaves  0  tens  ;  we  write  0  in  tens'  column  ;  etc. 

Abbreviated.  3,  0,  1.  We  look  upon  1  as  11  and  say  3 ;  we  look  upon 
0  as  9  and  say  0  ;  we  look  upon  5  as  4  and  say  1. 

Note.  —  Study  only  one  method. 


INTEGERS  31 

To  provey  add  the  subtrahend  and  the  remainder.     The  sum 
should  be  the  minuend.     See  p.  79. 

Subtract  and  prove,  explaining  infuU: 

12.  13.  14.  15. 

120      353      400      2000 
118      168      235      1234 


16. 

17. 

8000 

9000 

7001 

8023 

la 

19. 

20. 

21. 

22. 

23. 

320 

452 

500 

2001 

7000 

8306 

116 

348 

163 

1009 

6006 

7029 

Subtract  and  prove, 

abbreviating 

; 

24. 

25. 

2a 

27. 

2a 

29. 

500 

728 

514 

3000 

8003 

9750 

206 

599 

219 

1865 

7004 

7045 

30. 

31. 

32. 

3a 

34. 

35. 

782 

633 

708 

9028 

8705 

9454 

694 

238 
37. 

297 
3a 

8139 

7706 

7541 

36. 

39. 

40. 

41. 

613 

785 

308 

5106 

7006 

7755 

46 

98 

99 

987 

707 

794 

42. 

43. 

44. 

45. 

46. 

47. 

906 

555 

989 

4205 

7153 

8743 

807 

499 

893 

3986 

4154 

7139 

4a 

49. 

50. 

51. 

52. 

53. 

841 

736 

906 

8753 

2107 

5212 

693 

463 

809 

5645 

1009 

3271 

Ex.  24.  Ana.  204.  Say  4,  0,  2. 


82 


SUBTRACTION 


For  mental  and  written  work 

For  mental  practice^  nauie  each  figure  of  the  remainder  with- 
out writing  it,  and  without  stating  the  full  answer  after  it  is 
obtained. 


54. 
90543001 
76392132 

55. 

6398580044 
1296548980 

56. 

1094600105 
653864126 

57. 

75000S07063 
52983878156 

5a 

68686868 
16868698 

59. 

7899984343 
1549800022 

60. 

7205080901 
6789890993 

61. 

10340000900 
9876543211 

62. 

72980132 
68997654 

63. 

9453011342 
3568054321 

64. 

3900080040 
1777397786 

65. 

19876547001 
7687589422 

66. 

74840005 
69751126 

67. 

6365433309 
1350000677 

6a 

1000008000 
897329443 

69. 

18470020305 
17885332416 

7a 

63800402 
58911387 

6999214205 
5503436661 

72. 

9080706050 
4398616361 

73. 

97688888823 
88888888888 

74. 

54208075 
46677666 

75. 

3990911238 
1089832624 

76. 

7000000700 
6667893921 

77. 

78432234256 
7584476899 

78. 

90040708 
89753291 

79. 

6504003311 
1009876555 

80. 

6501030405 
5798398763 

81. 

10080090040 
9865732986 

Ex.  54.   For  mental  work,  say  9,  6,  8,  0,  6, 


INTEGERS  ,  88 

PROBLEMS 
Second  form  of  analysis 

82.  From  a  pile  of  184  apples,  96  apples  were  taken  away. 
How  many  remained? 

184, 

QQ  There  remained  the  difference  between  184 

apples  and  96  apples,  or  88  apples. 

88 

To  write  vohat  each  teitn  represents  is  of  service. 

83.  A  man  bought  a  horse  for  $  60  and  a  cow  for  $  24  less. 
What  was  the  cost  of  the  cow  ? 

^      *  Since  the  cow  cost  $  24  less  than  the  horse, 

^4t  «Wl^  less  she  cost  the  difference  between  $60  and  $24, 

d)/»  or  9  36. 

36f  cow  ^ 

A  drawing  is  often  of  service. 

84.  One  village  is  48  miles  west  and  another  37  miles  west  of 
Denver.     How  far  apart  are  the  villages  ? 


J&8    A  to  D  Since  it  is  48  miles  from  A  to  D,  and  37 

miles  from  B  to  D,  from  A  to  B  it  is  the 
^7y  B  to  D  difference,  or  11  miles. 

11,  AtoB 

85.  Which  number  is  nearer  to  912;  424  or  1200?    By  how 
luany  ? 

86.  A  house  was  sold  for  $2387,  or  for  $987  more  than  a 
fann.     What  was  the  selling  price  of  the  farm  ? 

87.  The  sum  of  two  numbers  is  980.  and  one  of  them  is  298. 
What  is  the  other  ? 

88.  The  difference  between  two  numbers  is  91)9,  and  the  greater 
is  1200.     What  is  the  smaller  ? 

AMkK.    AUllM. —  3 


34  ,  SUBTRACTION 

89.  A  man  bought  a  horse  for  $  270  and  sold  him  for  $  48  less. 
What  was  the  selling  price  ? 

90.  Counting  with  the  hands  of  a  watch,  how  many  minute 
spaces  are  there  from  the  point  at  VI.  to  the  point  at  VIII.  ? 
Draw  a  diagram. 

91.  From  a  farm  containing  1100  acres,  the  owner  sold  894 
acres.     How  many  acres  were  left  in  the  original  farm  ? 

92.  On  Monday,  Mr.  Willow  deposited  in  bank  $325;  on 
Tuesday,  $  87 ;  on  Wednesday,  $  87 ;  on  Thursday  he  drew  out 
$  255,  and  on  Friday,  $  44.     How  much  did  he  leave  on  deposit  ? 

9a  The  area  of  Texas  is  265,780  square  miles ;  Maine,  33,040 
square  miles;  New  York,  49,170  square  miles;  Pennsylvania, 
45,215  square  miles ;  Virginia,  42,450  square  miles ;  Massachusetts, 
8315  square  miles.  By  how  much  does  the  area  of  Texas  exceed 
the  combined  area  of  the  other  five  states  ? 

94.  A  steamer  sails  for  a  port  distant  2150  miles  ;  it  sails  195 
miles  on  Monday,  182  miles  on  Tuesday,  194  miles  on  Wednes- 
day, 188  miles  on  Thursday,  198  miles  on  Friday,  and  186  miles 
on  Saturday.     How  far  is  it  still  from  its  destination  ? 

95.  Three  j)ersons  buy  some  land  for  $  25,850 ;  the  first  pays 
$  9885,  the  second,  $  1025  more  than  the  first  How  much  does 
the  third  pay  ? 

9a  A  man  wishes  to  plant  450  trees ;  in  one  day  he  plants  46 
trees,  his  son  plants  38  trees,  and  each  of  three  men  who  are 
helping  him  plants  54  trees.  How  many  trees  remain  to  be 
planted  ? 

97.  A  gentleman  willed  $  98  more  to  each  of  two  sons  than  to 
each  daughter,  and  to  the  widow  $536  less  than  to  his  five 
children ;  each  son  received  $  866.  What  was  the  value  of  his 
estate? 

9a  Two  cities,  A  and  B,  are  896  miles  apart;  a  train  starts 
from  A  and  runs  258  miles  toward  B;  another  train  runs  188 
miles  from  B  toward  A.     How  far  apart  are  the  two  trains  ? 


MULTIPLICATION 


TERMS 


8  cents  =  4  cents  +  4  cents. 
This  is  commonly  written 

8  cents  =  4  cents  x  2. 
If  the  sum  is  wanting,  this  becomes 

what  =  4  cents  x  2  ? 
It  means  "  what  is  the  sum  when  4  cents  is  taken  two  times 
as  an  addend  ?  " 

Multiplication  is  the  process  of  find- 
ing the  sum  when  the  same  number  is 
used  several  times  as  an  addend. 

The  number  used  as  an  addend  is  the 
multiplicajid ;  the  number  showing  how 
many  times,  the  multiplier;  both  terms, 
factors;  the  result,  the  jyroduct. 

The  sign  of  multiplication  is  x . 

The  multiplier  may  precede  or  follow 
!lie  multiplicand.     In  the  former  case, 

X  *  is  read  times;  in  the  latter,  multi- 
plied by. 


Find  the  sum  when  8  is 
used  6  times  as  an  addend. 

8  +  8  +  8  +  8  +  8. 

8,  multiplicand, 
5,  multiplier. 
40j  product. 

2x4  cents, 

read 

£  times  4  cents. 


4  cents  x  2, 

read 

4  cents  multiplied  by  i. 


ADDITION   METHOD 

Multiplication  may  be  performed  by  addition. 

1.  In  this  way,  multiply  6  by  5. 

0x5=6  +  6  +  6  +  6  +  6;  adding,  we  obtain  30. 

2.  By  the  addition  method,  multiply  4  by  3;  multiply  3  by  4. 

36 


36  MULTIPLICATION 

COMMON   METHOD 

All  combinations  whose  products  are  less  than  100,  and  the 
multiplication  table  to  *12  times  12,'  should  be  memorized. 

3.  Memorize  the  following  : 

9  6  10  5  7  11  12  8  6  5  13 

2,3,  i<J;      2, 4,i?t?;    3,^i;     2^22;      2,3,4,^4;    5,25-,     2,26 

9  14  7  15  10  6  16  8  11  17  7 

3,^7;     2, 4,  ^<?;     2,    3,5,50;     2,4,5;?;     3,55;     2, 5-^;  6, 55 

18  12  9  6  19  13  20  10  8  21  14  7 

2,    3,4,6,55;      2, 55j       3, 5P;       2,    4,5,^;       2,    3,6,^ 

22  11  15  9  23  24  16  12  8  7  25  10 

2,  4,^;     3,5,^';     2,^6;     2,    3,    4, 6,  ^5;  7,  .^;     2,    5,50 

17  26  13  27  18  9  11  28  14  8  19 

3,  Ji;     2,    4,5^;      2,    3,6,5^;      5,55;     2,    4,7,55;     3,57 

29     30  20  15  12  10     31    21  9     32  16  8 
2, 5<9;   2,  3,  4,  5,  6,60;       2,62,    3,7,55;   2,  4,8,5^ 

13     33  22  11     34  17     23     35  14  10 
5,  55;   2,  3,  6,  66 \      2,  4,  55;   3,  5P;   2,  5,  7,  75 

36  24  18  12  8     37     25  15     38  19     11 
2,  3,  4,  6,9,7;?;   2,  7^;  3,  5,75;   2,  4,75;   7,77 

39  13  26      40  20.  16  10       27  9       41 
2,  6,  3,  75;    2,  4,  5,  8,  55;    3,  9,  5i;    2,  5^ 

42  28  21  14  12     17     43     29     44  22  11 

2,  3,  4,  6,  7,5^;  b,85.,     2,55;   3,57;   2,  4,  8,55 

45  30  18  15  10    13    46  23    31     47    19 
2,  3,  5,  6,  9,  P5;  7,  Pi;  2,  4,  P^;  3,55;   2,  P^;  b,  95 

48  32  24  16  12     49  14     33  11     50  25  20  10 
2,  3,  4,  6,  8,55;  2,  7,  P5;  3,  9,55;  2,  4,  5,  10,  i55. 

Thus :  9  twos,  6  threes,  18  ;  10  twos,  bfours,  20 ; .  .  .  . 


INTEGERS  37 

Declare  the  pwducts  rapidly  : 

4.  12,      5,      9,      4,    10,      6,    11,      2,      7,      3,      8,    by    IS. 

5.  4,  8,  11,  2,  7,  5,  10,  12,  3,  9,  6,  by  11. 

6.  8,  4,  6,  12,  2,  5,  10,  7,  11,  3,  9,  by  9. 

7.  6,  10,  2,  9,  12,  7,  4,  8,  3,  11,  5,  by  8. 
a  2,  7,  9,  5,  3,  11,  12,  8,  6,  10,  4,  by  7. 
9.  9,  5,  12,  8,  6,  3,  2,  10,  7,  4,  11,  by  6. 

10.  11,      4,      2,      9,    10,    12,      7,      5,      8,      6,      3,   by      5. 

11.  12,    10,      8,      6,      4,      2,    11,      9,      7,      5,      3,  by      4- 
Ex.  4.    144,  60,  108, ...     Do  not  say,  "  12  times  12  are  144." 

Declare  the  products  rapidly : 

12.  13  X  7,     13  X  3,     13  X  6,     13  x  2,     13  x  6,      13  x  4, 
14  X  6,     14  X  7,      14  X  3,      14  x  5,      14  x  2,      14  x  4. 

13.  15  X  4,     15  X  2,      15  X  5,     15  x  3,      15  x  6,      IG  x  4, 

16  X  6,    16  x  3,     16  X  2,     16  x  5,     17  x  3,     17  x  5,     17  x  2, 

17  x4. 

14.  18  X  3,     18  X  5,     18  X  2,      18  x  4,     19  x  3,     19  x  4, 
19  X  2,     19  X  5,     21  X  3,     21  x  2,     21  x  4. 

15.  22  X  2,     22  X  4,     22  x  3,     23  x  4,     23  x  2,     24  x  2, 
L'4  x  4,     24x3,     26x2,     25x4,     25x3. 

Ex.  12.   91,  30,  66,  .  .  .     Do  not  say,  ♦»  13  times  7  are  91." 

Declare  the  products  rapidly : 

16.  Of  4^  by  18,  2,  22,  6,  19.  3,  16,  21,  4,  24,  9, 16,  8,  20, 17,  6, 
23,  10,  13,  7,  11,  25, 14,  12. 

17.  Of  .<?,  by  30,  21,  17,  4,  9,  12,  20,  32,  29,  3,  23,  16,  2,  24,  6, 
11,  6,  22,  25,  7, 14,  26,  16,  27,  10,  28,  8,  13,  18,  31,  19. 

la   Of  i?,  by  50,  4,  9,  12,  4(5,  .36.  24,  11,  10,  20,  35,  46,  44,  34, 
22,  21,  9.  33,  7,  43,  32,  6,  42,  31,  19,  5,  30. 


88  MULTIPLICATION 

MULTIPLIER  LESS  THAN  THIRTEEN 
19.  Multiply  608  by  12  and  explain. 

S08  ^^  FULL.   12x8  units  are  96  units,  or  9  tens  and 

22  6  units ;  we  write  6  in  units'  column  and  carry  9 


tens. 


7296  12  times  0  tens  are  0  tens,  with  9  tens,  9  tens ; 

we  write  9  in  tens*  column. 
12  times  6  hundreds  are  72  hundreds,  or  7  thousands  and  2  hundreds ;  we 
write  2  in  Imndreds'  cohimn  and  7  in  thousands'  column. 

AnBREviATEi).     90,  9,  72. 

NoTK.  — It  is  a  great  loss  of  time  to  say  "  12  times  8  is  96;  12  times  0  is  0, 
and  {)  is  9."    While  these  words  are  being  said,  the  process  is  delayed. 

To  prove,  go  over  the  work  a  second  time.     See  p.  79. 
Multiply  and  prove,  explaining  in  full : 


20. 

21. 

22. 

23. 

24. 

25. 

633 
3 

234 
4 

944 
2 

7648 
2 

7651 
8 

7954 

8 

2& 

27. 

2a 

29. 

30. 

31. 

798 
6 

837 
9 

988 

7 

763 
5 

5494 
11 

6879 
12 

Multiply 

and  prove 

,  abbreviating 

: 

32. 

sa 

34. 

35. 

36. 

37. 

368 

7 

486 
9 

832 
8 

8497 
12 

2899 
11 

3312 
12 

3a 

39. 

40. 

41. 

42. 

43. 

437 
6 

902 

7 

350 

8 

2699 

7 

1848 
9 

4052 
5 

44. 

45. 

46. 

47. 

4a 

49. 

614 

7 

894 
8 

605 
12 

3424 
9 

1827 
2 

7320 
11 

Ex. 

32.  Ans, 

2576. 

Say  66,  47, 

25. 

INTEGERS 
For  mental  and  written  work 


89 


For  mental  practice,  name  each  partial  product  without  writing 
it,  and  without  stating  the  full  answer  after  it  is  obtained. 


5a 

90876514 
9 

51. 

3203164032 
8 

52. 

1543827006 
8 

53. 

25196078934 

7 

54. 
50247638 
5 

55. 

8223401523 
12 

56. 

5721604328 
6 

57. 

21398765430 

8 

5a 

25306801 
11 

59. 

1643525431 
9 

60. 

1704008006 
12 

61. 

52768430807 
11 

62. 

44667788 
12 

63. 

9364531046 
4 

64. 

0843086374 
9 

65. 

05280076403 
11 

66. 

30604781 
8 

67. 

9780365415 

7 

6a 

5943240826 
5 

69. 
06672884732 
12 

70. 

26082654 
8 

71. 

4932784609 
9 

72. 

8269000365 

7 

7a 

20060800072 
8 

74. 
92074031 
9 

75. 
4693278427 
12 

76. 
8327936002 
8 

77. 

78405080329 
6 

Ex.  50.   For  mental  work,  say  36,  12,  46,  68,  ...  . 


40  MULTIPLICATION 

MULTIPLIER  GREATER  THAN   TWELVE 
7a  Multiply  308  by  624  and  explain. 


gQg  868  X  4  units   =     1472  units 

368  X  2  tens    =     736    tens 
868  X  6  hund.  =  2208      hund. 
368  X  624         =  229632  units 


624 


1472 
736 


2208  ^®  place  the  right-hand  figure  of  each  partial 

product  under  that  figure  of  the  multiplier  which 
produces  it. 


229632 

79.  Multiply  96  by  365. 

96 

365 
aiQf\  ^  ^h®  lower  number  contains  the  more  figures, 

•oi^  ^®  "^*y  ^®  ^^®  upper  number  as  the  multiplier. 

32<io 


35040 


80.  Multiply  9386  by  5006. 


46930 


9386 

5006  ^  there  are  ciphers  within  the  multiplier,  we 

ef*9if^  are  careful  to  place  the  right-hand  figure  of  each 

partial  product  under  the  figure  of  the  multiplier 

which  produces  it. 


46986316 


81.  Multiply  638  by  100. 

j^f^  If  the  multiplier  is  10,  we  annex  one  cipher  to 

the  multiplicand  ;  if  100,  two  ciphers ;  and  so  on. 


63800 


82.  Multiply  4300  by  230. 


4300 

2gQ  If  one  or  both  terms  end  with  a  cipher,   we 

neglect  the  ciphers  in  multiplying,  and  annex  to 
the  product  as  many  ciphers  as  have  been  neg- 
lected. 


129 

86 

989000 


INTEGERS 


41 


Perform  the  indicated  operation  : 
33.  7703  X  834.  90.  947G  x  876. 


84.  3769  X  235. 

85.  7777  X  864. 

86.  5896  X  338. 

87.  7832  X  985. 
8a  8965  X  801. 
89.  8299  X  624. 


91.  8972  X  911. 

92.  7009  X  861. 

93.  9866  X  706. 

94.  9006  X  807. 

95.  9763  X  491. 

96.  8008  X  909. 


97.  3452  X  953. 

9a  9047  X  162. 

99.  4210  X  444. 

100.  2875  X  523. 

101.  9532  X  231. 

102.  5231  X  215. 

103.  3261  X  635. 


Perform  the  indicated  operation : 

104.  7894  X  3400.         107.  2875  x  8200. 

105.  8261  X  7000.         loa  2863  x  1000. 

106.  3689  X  5900.         109.  3961  x  7200. 


110.  3250  X  6120. 

111.  5100  X  7400. 

112.  6200  X  2000. 


Perform  the  indicated  operation  : 

lia  686  X  876004.  123.  88888  x  4983.  133.  2334  x  45572. 

114.  984  X  100000.  124.  94689  x  2686.  134.  2345  x  18542. 

115.  876  X  900008.  125.  87634  x  6902.  135.  9347  x  52100. 

116.  666  X  693024.  126.  76894  x  8075.  136.  4741  x  47050. 

117.  801  X  597636.  127.  68977  x  9004.  137.  4361  x  49472. 
lia  943  X  787878.  12a  70007  x  7001.  13a  4282  x  53000. 

119.  496  X  840039.  129.  68947  x  8064.  139.  1125  x  97777. 

120.  776  X  880000.  130.  47899  x  7006.  140.  2207  x  39003. 

121.  998  X  432122.  131.  89724  x  4100.  141.  5903  x  39400. 

122.  259  X  820606.  132.  90876  x  7008.  142.  8345  x  12345. 


42  MULTIPLICATION 

PROBLEMS 
Third  form  of  analysis 

14a  If  there  are  88  books  iu  each  of  12  boxes,  how  many  books 
are  there  in  all  ? 

^^  Since  there  are  88  books  in  1  box,  in  12 

^^  boxes  there  are  12  times  88  books,  or  106fl 


1056  books. 

To  lorite  dejtniuly  ofUr  the  multiplicand  and  the  product  what  they  repre- 
9entt  L  of  service. 

144.  At  $  125  each,  how  much  will  56  horses  cost  ? 

fiSSf  cost  of  one 

56 
1fcf%  Since  1  horse  costs  %  125,  66  horses  will  cost 

^£^  66  times  $126,  or  $7000. 


1000,  cost  of  aU 

146.  A  man  bought  a  horse  for  $40  and  a  cow  for  twice  the 
cost  of  the  horse.     What  was  the  cost  of  both  ? 
f40t  cost  of  horse 

?  Since  the  cow  cost  twice  $40,  or  $80,  they 

80 f  cost  of  cow  both  cost  the  sum,  or  $  120. 

120y  cost  of  both 

A  drawing  is  often  of  service. 

146.  Three  villages  are  in  a  straight  line ;  A  is  27  miles  east 
of  Denver ;  B  is  3  times  as  far  west  of  Denver ;  and  C  is  west  of 
Denver  by  twice  the  distance  from  A  to  B.  How  far  is  it  from  A 
toC? 


D   A 


27, 

AtoD 

8h 

BtoD 

108, 

AtoB 

216, 

CtoD 

248, 

AtoG 

Since  it  is  27  miles  from  A  to  D,  from  B  to 
D  it  is  3  times  27  miles,  or  81  miles  ;  since  it  is 
27  miles  from  A  to  D,  and  81  miles  from  B  to 
D,  from  A  to  B  it  is  the  sum,  or  108  miles; 
etc. 


INTEGERS  48 

147.  Mr.  White  bought  216  tons  of  coal  at  $  14  a  ton.  How 
much  (lid  the  coal  cost  him  ? 

14a  A  sewing  machine  makes  419  stitches  a  minute.  How 
many  stitches  will  it  make  in  98,000  minutes  ? 

149.  A  man  bought  17  plows  at  $8  apiece,  14  cultivators  at 
$  1.3  apiece,  and  134  shovels  at  $2  apiece.  What  was  his  entire 
bill? 

150.  In  a  book  of  589  printed  pages,  there  are  32  lines  to  the 
page  and  12  letters  to  the  line.  How  many  letters  does  the  book 
contain  ? 

151.  Light  moves  185,172  miles  in  a  second,  and  passes  from 
the  sun  to  the  earth  in  493  seconds.  What  is  the  distance  from 
the  sun  to  the  earth  ? 

152.  It  is  estimated  that  the  Mississippi  river  deposits 
137,139,200  cubic  yards  of  solid  matter  in  the  Gulf  of  Mexico 
every  year.  How  many  cubic  yards  have  been  deposited  in  512 
years  ? 

153.  A  man  has  194  horses ;  the  average  worth  of  36  of  them 
is  $  86  a  head ;  of  82,  $  97  a  head ;  and  of  the  remainder,  $  78  a 
head.     What  is  the  value  of  the  whole  herd  ? 

154.  Mr.  Brown  sells  296  acres  of  land  at  $116  an  acre,  and 
invests  the  proceeds  in  9  city  lots  at  $3295  each.  How  much 
money  has  he  left  ? 

155.  A  took  a  railway  journey  of  93  miles ;  B  traveled  9  times 
as  far;  C,  12  times  as  far  as  A  and  B  together;  D,  13  times  as  far 
as  C  less  B.     How  many  miles  did  all  of  them  travel  ? 

156.  A  man  bought  7  sheep  at  $11  a  head;  twice  as  many 
cows  at  3  times  the  price  of  a  sheep ;  4  times  as  many  horses  as 
cows  at  5  times  the  cost  of  a  cow ;  and  enough  steers  to  make  100 
animals  in  all  at  the  difference  between  the  price  of  a  sheep  and 
of  a  cow.     How  much  did  he  pay  in  all  ? 


DIVISION 


TERMS 

8  cents  =  4  cents  x  2. 
If  the  multiplicand  is  wanting,  this  becomes  Equation  I ;  if  the 
multiplier  is  wanting,  Equation  II. 

8  cents  =  what  x  2.  (I) 

8  cents  =  4  cents  x  what  (II) 

These  are  commonly  written : 

what  =  8  cents  -^  2  ?  (I) 

what  =  8  cents  -^  4  cents  ?  (II) 

Equation  I  means  "what  is  the  other  of  two  numbers  when 

one  of  them  is  2,  and  their  product,  8  cents?"     Equation  II 

means  "  what  is  the  other  of  two  numbers  when  one  of  them  is 

4  cents  and  their  product,  8  cents  ?  " 


Division  is  the  process  of  finding  the 
other  of  two  numbers  when  one  of  them 
and  their  product  are  given. 

The  product  is  the  dividend;  the  num- 
ber given,  the  divisor;  the  number  re- 
quired, the  quotient. 

If  the  required  number  was  the  multipli- 
cand, division  becomes  the  process  of  find- 
ing one  of  the  equal  parts  into  which  a 
number  may  be  separated. 

If  the  required  number  was  the  multi- 
plier,  division  becomes  the  process  of  find- 
ing how  many  times  one  number  contains 
another. 

44 


One  of  two  numbers 
is  2,  and  their  product, 
8.     What  is  the  other  ? 

Ans.  4i  because  Bx4 
=  8.  8,  dividend;  S^ 
divisor;  -#,  quotient. 

What  is  one  of  the 
two  equal  parts  into 
which  §  8  may  be  sepa- 
rated ? 

Ans.  f  ^,  because  f8 
=  f4xe,ov  f4  +  f4' 

How  many  times  does 
$  8  contain  $  4  ? 

Ans.  2  times,  because 
f  8=2x§4' 


INTEGERS 


46 


Division  is  expressed  in  four  ways : 

By  writing  the  dividend  above  and  the 
divisor  below  a  horizontal  lirie. 

By  writing  the  sign  *  -5- '  betiveen  the 
tenns. 

By  writing  the  sign  ' : '  between  the 
tmns. 

By  icriting  the  dividend  at  the  right 
and  the  divisor  at  the  left  of  a  curved  line. 


Q 

-,  fractional  method. 


8  -i-  S^  common  method. 
8 :  f ,  ratio  method. 
i)8y  working  method. 


ADDITION  METHOD 
Division  may  be  performed  by  addition. 

1.  In  this  way,  divide  8  cents  by  2. 

8  cents  -h  g  calls  for  the  other  of  two  numbers  when  one  of  them  is  2,  and 
their  product,  8  cents ;  or  it  calls  for  the  number  that  will  produce  8  cents 
when  taken  2  times  as  an  addend.  1  cent  taken  2  times  as  an  addend  pro- 
duces 1  cent  -f  1  cent,  or  2  cents ;  £  ce7Us,  2  times  as  an  addend,  2  cents  + 
2  cents,  or  4  cents ;  5  cents,  2  times  as  an  addend,  3  cents  +  3  cents,  or  6 
cents  ;  4  cents,  2  times  as  an  addend,  4  cents  +  4  cents,  or  8  cents.  There- 
fore, 8  cents  -i-  £  =  4  cents. 

2.  In  this  way,  divide  8  cents  by  4  cents. 

8  cents  -r-  4  cents  calls  for  the  other  of  two  numbers  when  one  of  them  is 
4  cents,  and  their  product,  8  cents ;  or  it  calls  for  the  number  of  times  that  4 
cents  must  be  used  as  an  addend  to  produce  8  cents.  4  cents  used  as  an 
addend  once  produces  4  cents ;  4  cents,  as  an  addend  2  times,  4  cents  +  4 
cents,  or  8  cents.    Therefore,  8  cents  -r-  4  cents  —  S. 

3.  By  the  addition  method,  divide  15  quarts  by  3;  explain  in 
full. 

4.  By  the  addition  method,  divide  15  quarts  by  5  quarts; 
explain  in  full. 

5.  By  the  addition  method,  divide  16  qtiarts  by  4  quarts; 
explain  in  full. 


46  DIVISION 

COMMON   METHOD 

If  the  combinations  in  multiplication  have  l>een  mastered,  the 
results  in  the  following  examples  may  be  called  rapidly.  In 
multiplication,  we  have  two  factors  to  name  their  product;  in 
division,  the  product  and  one  factor  to  name  the  other. 

6.    Oiven  the  product  and  one  factory  name  the  other : 

9  14  7  15  10  6  16  8  11  17  7 

?,  j?7;    ?,  ?,  iS*;     ?,    ?,  ?,  ^0;     ?,  ?,  5^;     ?,  5^;     ?,  ^^;    ?,  5^; 

18  12  9  6  19  13  20   10  8  21    14  7 

?,    ?,?,?,  ^6;       ?,A9j       ?,5P;       ?,     ?,?,^(?;       ?,     ?,  ?,  ^^ 

22  11  15  9  23  24  16  12  8  7  25  10 

?,    ?,^;     ?,?,^;     ?,4^;     ?,    ?,    ?,  ?,  ^;    ?,-^5;     ?,    ?,  5(?; 

17  26  13  27  18  9  11  28    14  8  19 

?,5i;      ?,    ?,5je;      ?,    ?,?,5^;      ?,55;      ?,     ?,?,^e;      ?,57; 

29  30  20  15  12  10  31  21  9  32  16  8 

?,^;   ?,  ?,  ?,  ?,  ?,^^;  ?,^^;  ?,?,^^;  ?,  ?,  ?,64; 

13      33  22  11      34  17      23      35  14  10 
?,^5;    ?,  ?,  ?,5^;    ?,  ?,6<y;    ?,6P;    ?,  ?,  ?,  7t?; 

36  24  18  12  9        37        25  15        38  19 

?,  ?,  ?,  ?,?,7;?;      ?,7-^;      ?,  ?,  7J;      ?,  ?,  7^; 

11      39  13      40  20  16  10       27  9       41 
"^,77-,        ?,  ?,7<J;    ?,  ?,  ?,  ?,<90;    ?,?,<^i;    ?,<^^; 

42  28  21  14  12      17      43      29      44  22  11 

?,  ?,  ?,  ?,  ?,,J4;   ?,<^^;   ?,<^^;   ?,<^7';   ?,  ?,  ?,<?<y; 

45  30  18  15  10     13     46  23     31    47     19 
?,  ?,  ?,  ?,  ?,P0;  ?,5i;  ?,  ?,  P^;  ?,P5;  ?,^4;  ?,  ^5; 

48  32  24  16  12     49  14     33  11     50  25  20  10 
?,  ?,  ?,  ?,  ?,  P<5;   ?,  ?,  P<9;   ?,  ?,  PP;   ?,  ?,  ?,  ?,  i^^ 

Thus :  3 ;  2,  4  J  2,  3,  5  ;  ...  . 


INTEGERS  47 

Declare  the  quotients  rapidly  : 

7.  144,  48,  96,  36,  120,  72,  24,  132,  84,  108,  60,  ^  12. 

a  99,  27,  54,  90,  108,  18,  63,  81,  36,  72,  45,  -s-  0. 

9.  72,  48,  24,  96,  16,  80,  40,  32,  88,  56,  64,  -  8. 

10.  70,  14,  49,  63,  77,  5%,  28,  84,  21,  42,  35,  --  7. 

11.  24,  42,  72,  54,  36,  60,  12,  66,  30,  18,  48,  ^  6. 

12.  50,  10,  20,  30,  40,  60,  45,  55,  35,  15,  25,  h-  5. 

13.  44,  28,  20,  48,  12,  32,  40,  36,  16,  24,  8,  --  4. 

14.  27,  18,  6,  36,  24,  9,  33,  21,  12,  30,  15,  --  8. 

Ex.  7.   12,  4,  8,  3,  10,  ...  . 

Declare  the  quotients  rapidly : 

15.  98-^49,  98-5-14,  96-24,  96 -f- 16,  95-^19,  94h-47, 
92H-23,    91-i-13,    90H-18,    90^30,    88^22. 

16.  87-5-29,  86-5-43,  84-5-28,  84-5-14,  82-5-41,  80 -5- 16, 
78H-13,   76^-19,   75-5-15,   74-5-37,   72-5-24,  72-4-36,  72 -- 18. 

17.  70^14,  68^-17,  66^22,  65-5-13,  64-^16,  63-5-21, 
62-31,    60-hl5,    68-*-29,    57-5-19,    56-5-14,    54-18. 

la  52-5-13,    51-4-17,    48-16,    48-^24,    46-23,    45-5-15, 
42-5-14,    39-5-13,    38-5-19,    34-^17,    32 -h  16,    28-5-14. 
Ex.  15.  2,  7,  4,  6, 

Dedare  the  quotients  rapidly  : 

19.  24:12,  34:17,  48:8,  27:9,  35:7,  39:13,  45:9, 
26:13,     42:14,     60:12,     44:11,     33:11. 

2a  42:21,  32:16,  42:21,  45:15,  84:12,  84:21,  84:42, 
96:12,    36:18,    66:6,     66:11,     68:4. 

21.  72:18,  38:19,  30:15,  75:5,  75:16,  90:18,  68:17, 
95:19,     61:17,     48:16,     84:6,     98:14. 

Ex.  19.   2,  2,  6,  8,  5, 


48  DIVISION 

The  remainder 

The  dividend  is  not  always  the  product  of  the  divisor  and  an 
integer. 

In  this  case,  the  largest  integral  quotient  is  found,  and  the  re- 
mainder, obtained  by  subtracting  the  product  from  the  dividend, 
is  left  undivided.  The  remainder  is  usually  written  above  and 
the  divisor  below  a  horizontal  line  to  show  that  the  division  has 
not  been  performed.     Thus : 

In  9  -^  2,  the  largest  integral  quotient  is  4,  because  2x4  =8,  and  8  from 
9  leaves  1,  a  remainder  smaller  than  the  divisor.  Therefore  9-4-2  =  4  with  a 
remainder,  1,  which  is  left  undivided  ;  or,  9  -=-  2  =  4J  ;  read,  9  -f-  2  =  4  and 
1  -i-  2,  or  4  and  one  half. 

Find  the  value  of: 

22.  19^3,  17-i-4,  20  +  3,  21 --2,  ll-j-3,  17 --2,  26  +  5, 
29  +  3,  33  +  4,   27  +  6,   43  +  7,   31  +  8,   15  +  7,  23  +  7. 

2a  16  +  6,  27  +  4,  37  +  4,  29  +  3,  39  +  2,  38  +  3,  19  +  8, 
43^8,    26  +  3,    35  +  6,    37  +  3,    29  +  7,    41+5,    27  +  5. 

24.  22  +  5,  38  +  8,  46  +  5,  36  +  7,  48  +  7,  41  +  6,  49  +  6, 
33^7,    42  +  5,    36  +  7,    51  +  5,    47  +  9,    53  +  7,    42  +  9. 

25.  21+6,  26  +  7,  34  +  4,  36  +  5,  49  +  3,  51  +  2,  43+3, 
46^7,    61  +  3,    22  +  7,    46+3,    28  +  8,    39  +  7,    51  +  7. 

Ex.  22.   6,  1  ;  4,  1 ;  6,  2  ;  10,  1 ;  3,  2 ; 

Find  the  value  of: 

26.  119,   17,   111,   113,     14,   117,  86,  118,  90,  109,  91,  + 12. 

27.  65,  97,     45,   110,     78,     56,  32,    75,  23,    85,  98,  + 12. 
2a   107,   68,     73,     95,     25,     59,  64,    28,  35,  102,  80,  + 11. 

29.  20,   50,     40,     70,   100,     60,  30,  90,  85,  63,  79,  + 11. 

30.  89,   58,     26,     67,     39,     75,  16,  32,  48,  83,  60,  +    9. 

31.  15,   34,     55,     70,     80,     20,  42,  64,  86,  82,  75,  +  9. 
Ex.  26.    9,  11  ;  1,  5 ;  9,  3  ;  9,  5  ;  1,  2  ; 


INTEGERS  49 

Equal  parts 

A  whole  may  be  separated  iuto  equal  parts. 

When  the  whole  is  separated  into  two  equal  parts,  each  part  is 

a  ImL/;  into  three  equal  parts,  each  part  is  a  third;  into  four 

equal  parts,  each  part  is  a  quarter,  or  a.  fourth;  and  so  on. 

Since  8  =  4  +  4,  one  halfoi  8  is  4.    This  is  written  ^  of  8  =  4. 
Since  6  =  2  +  2  +  2,  one  third  of  6  is  2.     This  is  written  ^  of  6  =  2. 

Dividing  by  2  is  finding  a  half,  dividing  by  3  is  finding  a  third, 

dividing  by  4  is  finding  a  fourth,  and  so  on.     Thus : 

8  cents  ^2=4  cents,  because  8  cents  =  4  cents  x  2. 

^  of  8  cents  =  4  cents,  because  8  cents  =  4  cents  +  4  cento. 

Find  the  value  of  one  part  : 

32.  When  12^  is  separated  into  4  equal  parts ;  18^  into  3  equal 
parts. 

33.  When  20^  is  separated  into  5  equal  parts ;  28^  into  4  equal 
parts. 

34.  When  60^  is  separated  into  6  equal  parts ;  72^  into  8  equal 
parts. 

35.  When  45^  is  separated  into  9  equal  parts ;  50^  into  5  equal 
parts. 

36.  When  35^  is  separated  into  7  equal  parts ;  45^  into  9  equal 
parts. 

Kx.  32.    3^.     Zf  +  3)^  +  3)»  +  3/»  =  \2f. 

Find  the  value  of: 

37.  i  of  $6;     1  of  $  12.  42.  \  of  96^;  J  of  72^. 
3a   J  of  «20;  i  of  «30.  43.   J  of  80^;  -jir  of  66^. 

39.  i  of  $12;  \  of  «56.  44.  \  of  60^;  ^  of  91^. 

40.  iof  $46;  Jof  $72.  45.  jof84^;  J  of  64^. 

41.  i  of  $81 ;  J  of  $56.  46.  1  of  90^;  J  of  66^. 

Ex.  37.  $3.     I  of  f  6  =  f  6  +  2,  or  f  3. 

NoTB.  —  The  examples  on  pp.  47  and  48  should  be  reviewed  as  follows :  Ex.  7, 
p.  47.    Aof  144  is  12;  of  48,  4; Ex.  2(>,  p.48.    i^ofll9U9}i; 

▲  MBR.    ARITH.  —  4 


60  DIVISION 

Times  contained 
A  whole  may  contain  a  part  an  exact  number  of  times. 

A  whole  contains  its  half  two  times ;  its  third,  three  times ;  its 
fourth,  four  times ;  and  so  on.  12  cents  contains  6  cents  2 
times ;  4  cents,  3  times ;  and  so  on.     Thus : 

8  =  4  +  4  ;  or  8  =  i  of  8  +  i  of  8. 

6  =  2  +  2  +  2  ;  or  6  =  J  of  6  +  }  of  6  +  i  o'  ^• 

Dividing  a  number  of  cents  by  4  cents  is  finding  how  many 
times  the  number  contains  4  cents ;  dividing  a  number  of  eggs  by 
5  eggs  is  finding  how  many  times  the  number  contains  5  eggs ; 
and  so  on.     Thus : 

8  cenU  -i-  4  cents  =  2,  because  8  cento  =  4  cento  x  2. 

8  cento  contains  4  cento  2  times,  because  8  cento  =  4  cento  x  2. 

How  many  times  does : 

47.  26  days  contain  13  days  ?  24  days  contain  12  days  ? 
4a   20  hours  contain  2  hours  ?  16  hours  contain  4  hours  ? 
49.  18  gallons  contain  2  gallons  ?  96  gallons  contain  6  gallons  ? 
5a  25  quarts  contain  5  quarts  ?  60  quarts  contain  4  quarts  ? 
Ex.  47.  2  times.    26  days  =  13  days  x  2. 

How  many  times  is  : 

51.  2  pints  contained  in  18  pints  ?  3  pints  in  12  pints  ? 

52.  4  pecks  contained  in  20  pecks  ?  5  pecks  in  30  pecks  ? 

53.  3  pounds  contained  in  15  pounds  ?  6  pounds  in  72  pounds  ? 

54.  5  ounces  contained  in  30  ounces  ?  7  ounces  in  42  ounces  ? 

Ex.  51.    9  times.     18  pinto  =  2  pinto  x  9. 

Note.  — The  examples  on  pp.  47  and  48  should  be  reviewed  as  follows:  Ex.  7, 
p.  47.  144  contains  12, 12  times ;  .  .  .  .  Ex.  26,  p.  48.  119  contains  12,  9  times 
with  11  remaining. 


INTEGERS  51 

SHORT  DIVISION 

55.  Divide  8609  by  12  and  explain. 

In  full.  86  Imndred8-f-12=7  hundreds  and  2  hun- 
12)  8609  dreds  remaining  ;  we  write  7  in  hundreds'  cohinin. 

717I.  2  hundreds  and  0  tens  =  20  tens  ;   20  tens  -=-  12 

It  =1  ten  and  8  tens  remaining ;  we  write  1  in  tens' 

column. 
8  tens  and  9  units  =  80  units  ;  89  units  -r- 12  =  7  units  and  5  units  remain- 
ing ;  we  write  7  in  units'  column  and  12  under  6,  with  a  line  between,  to 
show  that  6  is  still  to  be  divided  by  12. 
Abbreviated.   7,  1,  7,  ^j. 

Note.  —  It  is  a  great  loss  of  time  to  say,  "  86  divided  by  12  is  7  with  2  remain- 
ing."   While  these  words  are  being  said,  the  process  is  delayed. 

To  provey  multiply  the  divisor  by  the  quotient  and   add  the 
remainder.     The  result  should  be  the  dividend.     See  p.  79. 

Divide  and  prove,  explainitig  in  fuU : 


5e 

57. 

58. 

59. 

60. 

2)864 

3)981 

9)648 

8)1238 

12)9754 

61. 

62. 

6a 

64. 

65. 

9)369 

11)858 

7)504 

6)3255 

4)2367 

Divide  and 

prove,  abbreviating  : 

66. 

67. 

6a 

69. 

70. 

9)495 

6)558 

8)696 

7)3728 

5)4163 

71. 

72. 

73. 

74. 

75. 

12)864 

9)718 

7)623 

8)3001 

12)7000 

76. 

77. 

7a 

79. 

80. 

11)979 

6)425 

4)916 

3)7070 

2)5007 

Bx.  66.   Ana.  66.    Say  6,  6. 


52 

] 

DIVISION 

For  mental  and  written  work 

For  mental  practice,  name  each  figure  of  the  quotient  without 
writing  it,  and  without  stating  the  full  answer  after  it  is  obtained. 

81. 

12)567024 

82. 

9)3063205 

83. 

11)30670508 

84. 

9)1023456789 

85. 

11)781605 

86. 

7)3643036 

87. 

9)20345607 

88 

8)3321456648 

89. 

9)612036 

90. 

5)3245321 

91. 

8)72867408 

92. 

7)1111111111 

9a 

8)921608 

94. 
6)3215750 

95. 
7)30000005 

96. 

6)7340962416 

97. 

7)333333 

98. 

3)2222667 

99. 
6)55555554 

loa 

5)9765432120 

Id. 

6)100308 

102. 

6)5307943 

loa 

5)98765043 

104. 

4)9998887772 

105. 

5)666665 

106. 

4)5202415 

107. 

4)12345672 

loa 

3)9012506370 

109. 

4)999904 

110. 

9)1396735 

111. 

3)12345672 

112. 

2)9012506370 

113. 

3)444444 

114. 

7)5803214 

115. 

2)56708914 

116. 

12)9876954312 

117. 

2)973514 

iia 

8)5905309 

119. 

12)63178908 

120. 

11)5443322344 

Ex.  81.    For  mental  work,  say  4,  7,  2,  5,  2. 


INTEGERS  53 

LONG  DIVISION 

121.  Divide  8609  by  12  and  explain. 

717-^  86  hundreds  -^12  =  7  hundreds  and  2  hundreds 

i&\o^nq~  remaining ;  we  write  7  in  hundreds'  column. 

8 A  ^  hundreds  and  0  tens  =  20  tens  ;  20  tens  -  12 

— '^—  =  1  ten  and  8  tens  remaining ;  we  write  1  in  tens' 

^0  column. 

1^  8  tens  and  9  units  =  89  units ;  89  units  -r- 12 

S9  =7  units  and  6  units  remaining ;  we  write  7  in 

Sj^  units'  column  and  12  under  5  with  a  line  between, 

~e  to  show  that  6  is  still  to  be  divided  by  12. 

122.  Divide  34056  by  17,  prove,  and  explain. 

^  i7  34  thousands  -=-  17  =  2  thousands  and  0  thou- 

17)  S4O66  sands   remaining ;    we    write   2    in    thousands' 

S4  column. 

0  hundreds  -4-  17  =  0  hundreds  ;  we  write  0  in 
hundreds'  column. 

6  tens  -f- 17  =  0  tens  and  5  tens  remaining ;  we 

write  0  in  tens'  column. 

Proof.  5  tens  and  6  units  =  56  units  ;  56  units -i- 17=3 

2003—  units  and  6  units  remaining ;  we  write  3  in  units' 

jrj^^  column  and  17  under  5  with  a  line  between,  to 

z —  show  that  5  is  still  to  be  divided  by  17. 

14021  Proof.    Multiplying  the  divisor  by  the  quotient 

2003  mid  adding  the  remainder,  we  obtain  the  dividend. 

34056 

Dividej  provcj  explain : 

7649 -t- 21;    4179  +  22.  12a   30000 -f- 16;    6875  +  16. 


0066 

61 

6 


124.  3174  +  23;  8754  +  24.  129.  85038  +  17;  6283  +  17. 

125.  2800  +  13;  6200  +  13.  130.  72004  +  18;  5796  +  18. 

126.  9963  +  14;  3528+14.  131.  38794  +  19;  4986  +  19. 

127.  8072  +  15;  6123  +  15.  132.  79498  +  25;  3899  +  23. 


54  DIVISION 

133.  Divide  896  by  112. 

8  To  find  the  quotient  figure  when  the  first 

jyo\QQ^  figure  of  the  dividend  is  larger  than  the  first 

HSS)<5ifo  figure  of  the  divisor,  we  use  only  the  first  figure 

896  of  each  term. 

Approximately,  896  -»- 112  =  8  -j- 1,  or  8.    8  x  112  =  896. 

134.  Divide  2992  by  374. 

8  To  find  the  quotient  figure  when  the  first 

figure  of  the  dividend  is  smaller  than  the  first 

874)2992  figure  of  the  divisor,  we  use  only  the  first  two 
2992  figures  of  the  dividend  and  the  first  figure  of 
the  divisor. 

Approximately,  2992  +  374  =  29  h-  3,  or  9.     9  x  374  =  3366.    Since  9  is 
too  large,  we  try  8. 

135.  Divide  3808  by  476. 

8 


To  determine  whether  the  quotient  figure  is 
476)3808  too  lai^e,  it  is  rarely  necessary  to  use  more 

S808  ^^""1  ^6  fi^^^  ^^o  figures  of  the  divisor. 


Approximately,  3808  ^  476  =  38  -j-  4,  or  9.    9  x  47  =  423.    Since  9  is  too 
large,  we  try  8. 

Perfonn  the  indicated  operation  : 

136.  730-^365.         141.  8300^426.  146.  16748 -- 4187. 

137.  980-245.    142.  8502-5-321.  147.  10380 -- 2076. 
13a  845-169.    143.  7321-375.  14a  47472-5934. 

139.  984-5- 123.    144.  6325-523.  149.  35315 -r- 7063. 

140.  981-5-109.    145.  8053^437.  150.  61263-5-6807. 


INTEGKKS 


66 


151.  Divide  78264648  by  98076. 


98076)7SiS6464S(798 
686532 

961144 
882684 


784608 
784608 


78-5-9  =  8,  8x98  =  784.  Since  8  is  too 
large,  we  try  7. 

Since  the  product  of  98076  by  7  must  con- 
tain six  figures,  we  place  2,  the  right-hand 
figure  of  the  product,  under  the  sixth  figure 
of  the  dividend. 


Note.  —  It  is  sometimes  more  convenient  to  place  the  quotient  at  the  right. 


152.   Divide  7384  by  100. 
100)7384 


7^lk 


If  the  divisor  is  10,  we  cut  off  one  figure 
from  the  right  of  the  dividend ;  if  100,  two 
figures ;  and  so  on. 


153.  Divide  3218738  by  92000. 


92.000^)3218.738^(34^ 
276 
458 
368 


90 


If  the  divisor  ends  with  ciphers,  we  cut 
off  the  ciphers,  and  the  same  number  of 
figures  from  the  right  of  the  divisor. 

We  divide  the  parts  left  and  prefix  the 
remainder  to  the  part  cut  off,  to  find  the 
true  remainder. 


NoTB.  —  A  cross  may  be  placed  after  units,  and  a  point  before  the  last  figure 
cut  off. 

154.  Prove  the  answer  of  the  last  example. 


<a,90TS8 

92000 
90738 
68 

306 

3218738 


Multiplying  the  quotient  by  the  divisor 
and  adding  the  remainder,  we  obtain  the 
dividend. 


Find  the  value  of: 

155.  1468953  +  47963. 

156.  9547964  +  5432a 


157.  8197385 -J- 38900. 
15a  4178967+46000. 


56 


DIVISION 


Perform  the  indicated  operation : 

159.  106066  +  34. 

160.  126499 -f- 29. 
Id.  172929-1-69. 

162.  209676-!- 68. 

163.  381961-!- 93. 

164.  196600 -i- 76. 

165.  336777-*- 87. 

Perform  the  indicated  operation  : 

173.  476204-!- 298. 

174.  978383-^487. 

175.  776984-^-964. 

176.  328372 -^  878. 

177.  376624 -^  888. 
17a  389961  +  117. 
179.  904321-!- 229. 

Perform  the  indicated  operation : 

187.  468797 -i- 1000. 

laa  776936-!- 2000. 

189.  589476 -^  2200. 

190.  987664  +  4600. 

191.  234869  +  9100. 

192.  489763  +  8000. 

193.  940849  +  7000. 


166.  1344466  +  36. 

167.  1332678+46. 
168  1670688  +  64. 

169.  3334932  +  73. 

170.  2646660  +  62. 

171.  7920792  +  88. 

172.  6141408  +  76. 

180.  3032676  +  97826. 

181.  8964200  +  44821. 

182.  9777680  +  44444. 

183.  7642639  +  78788. 

184.  9916620  +  94444. 

185.  2661733  +  10809. 

186.  3649780  +  78884. 


194.  9285564 

195.  8609250 

196.  8940008 

197.  9486534 
198  4394988 

199.  4333333 

200.  2340899 


-29292. 
^68866. 
-41000. 
■r-  19900. 
4- 14000. 
■f- 34000. 
-  20000. 


INTEGERS  67 

PROBLEMS 
Fourth  and  fifth  forms  of  analysis 
2aL   If  48  oranges  cost  96^,  how  much  will  1  orange  cost? 

__? 
4S)  96  Since  48  oranges  cost  96^,  1  orange  will  cost  ^ 

96  oi9Qf,OT2f. 

Proof.  4Sx  gf=i  96 f. 

202.  At  2^  each,  how  many  oranges  can  be  bought  for  96^. 

^)96  Since  1  orange  costs  2  ^,  as  many  oranges  can 

4S  be  bought  for  96  ^,  as  2  ^  is  contained  times  in  96  ^, 

Proof.    gfx48=:9€f.     or 48  oranges. 

To  write  what  each  term  represents  is  of  service. 

203.  If  17  acres  of  land  cost  $204,  how  much  will  1  acre  cost? 
f  12,  cost  1  acre 

17)20J^y  cost  all 

-^^  Since  17  acres  cost  ^204,  1  acre  will  cost  ^  of 

34  «204,  or^l2. 

Ji 

Proof.   17y.%12=%i0j^ 

A  drawing  is  often  of  service. 

20i.  Three  villages  are  in  a  straight  line ;  B  is  3  times  as  far 
west  as  A  is  east  of  Denver ;  C  is  twice  as  far  west  of  B  as  B  is 
west  of  Denver ;  from  B  to  C  is  18  miles.  How  far  is  it  from  A 
toC? 

C  B  D  A 


Since  2  times  BD  =  18  miles,  BD  =  18 
18,  B  to  C  m.!e8  -=-  2,  or  9  miles.     Since  3  times  DA  =  9 

9,  B  to  D  miles,  DA  =  9  miles  -h  3,  or  3  miles.     Dis. 

s]  D  to  A  ^"^®  AC=  DA+  BD+  BCy  or  3  miles  +  9 

-Tz*   ^  .    ri  miles  +  18  miles,  or  30  miles. 


58  DIVISION 

205.  If  the  cost  of  constructing  459  miles  of  railway  Ms 
$596,700,  what  is  the  cost  per  mile? 

206.  If  1,500,000  people  occupy  a  territory  of  25,000  square 
miles,  what  is  the  average  population  on  each  square  mile  ? 

207.  If  324  acres  of  land  produce  20,736  bushels  of  corn,  what 
is  the  average  yield  per  acre? 

20a  The  salary  of  the  President  of  the  United  States  is 
f  50,(X)0.  How  much  is  that  a  day,  counting  365  days  to  the 
year  ? 

209.  My  front  fence  is  5  rods  long  and  cost  $  130.  How  much 
did  it  cost  per  rod  ? 

210.  Six  men  owning  a  mine,  sold  it  to  11  others  for  $  33,000. 
How  much  did  each  of  the  original  owners  receive,  and  how 
much  did  each  of  the  new  owners  pay  ? 

211.  If  a  man  pays  $  16  rent  per  month,  in  how  many  years 
of  12  months  each  will  he  pay  $  1152  rent? 

212.  In  how  many  days  will  a  cooper  make  1356  barrels,  if  he 
makes  12  barrels  each  day  ? 

213.  A  miller  packed  26,950  pounds  of  flour  into  sacks  contain- 
ing 49  pounds  each.     How  many  sacks  did  he  fill  ? 

214.  There  are  3  feet  in  a  yard  and  1760  yards  in  a  mile.  How 
many  miles  are  there  in  63,360  feet  ? 

215.  Into  how  many  farms  of  160  acres  each  can  900  acres  of 
land  be  divided,  and  how  many  acres  will  remain  ? 

216.  After  dividing  900  acres  of  land  into  farms  of  160  acres 
each,  a  man  sold  what  was  left  for  $200.  How  much  did  he 
receive  per  acre  ? 

217.  How  many  pages  are  there  in  a  book  containing  87,024 
words,  if  each  page  contains  37  lines,  and  each  line  contains  14 
words  ? 

2ia  Mr.  Gray  leases  a  house  for  $  27  a  month.  If  the  expenses 
are  $8  a  month,  in  how  many  months  will  he  gain  $6016  from 
the  house? 


INTEGERS  59 

219.  A  man  had  $386,  which,  lacking  $5,  was  17  times  as 
much  as  he  had  10  years  ago.  How  much  was  he  worth  10  years 
ago? 

22a  The  area  of  Kansas  is  82,080  square  miles ;  of  Rhode 
Island,  1250  square  miles.  The  area  of  Kansas  is  approximately 
how  many  times  that  of  Rhode  Island  ? 

221.  If  I  subtract  the  product  of  375  and  25  from  the  product 
of  675  and  39,  and  divide  the  remainder  by  75,  what  is  the 
quotient  ? 

222.  A  man  sells  2  houses  at  $3315  apiece,  and  with  the  pro- 
ceeds buys  land  at  $  65  an  acre.     How  many  acres  does  he  buy  ? 

223.  A  merchant  paid  $817  for  19  stoves,  and  afterwards  sold 
them  for  $51  each.  How  much  did  the  selling  price  of  all  ex- 
ceed the  cost  ? 

224.  How  many  tons  of  hay  at  $  13  a  ton,  the  cost  of  shipping 
being  $  2  a  ton,  can  a  man  buy  in  exchange  for  16  horses  at  $  153 
a  head,  the  cost  of  delivery  being  $3  for  each  horse,  if  he  pays  for 
both  shipping  and  delivery  ? 

225.  A  merchant  receives  as  much  for  65  pounds  of  butter  as 
for  15  yards  of  cloth  at  78  cents  a  yard.  What  is  the  price  of  the 
butter  per  pound  ? 

226.  A  boy  makes  a  journey  of  16  hours  by  rail,  traveling  32 
miles  an  hour;  he  returns  on  a  bicycle  at  the  rate  of  8  miles  an 
hour.     How  long  is  he  in  returning  ? 

227.  John  and  James  run  a  race  of  720  feet;  John  runs  16  feet 
in  a  second  apd  beats  James  by  3  seconds.  How  many  feet  does 
James  run  in  a  second  ? 


OPERATIONS  COMBINED 


USE  OF  THE   SIGNS 


The  whole  is  equal  to  the  sum  of  all 
its  parts ;  or  the  whole,  diminished  by  all 
its  parts,  becomes  nothing. 

The  signs  *-h'  and  *— *  perform  the 
double  office  of  connecting  parts  and  de- 
noting operations.  Whatever  is  included 
between  the  signs  *-f  *  or  *— '  is  a  term. 
The  terras  to  the  left  of  the  sign  *=* 
form  the  left-liand  member ;  the  terms 
to  the  right,  the  right-hand  m,ember ; 
both  members,  an  equation. 

A  simple  term  may  be  reduced  to  a 
compound  term  by  expressing  it  as  a 
sum,  a  difference,  a  product,  or  a  quo- 
tient. 

When  a  compound  term  is  a  sum  or  a 
difference,  its  components  are  connected 
by  the  signs  ^  +  '  or  *— ',  and  must  there- 
fore be  bound  together  by  a  parenthesis, 
by  brackets,  or  by  a  vinculum. 

When  a  compound  term  is  a  product 
or  a  quotient,  its  components  are  not 
connected  by  the  signs  *  +  ^  or  *  — ',  and 
a  parenthesis  is  not  needed. 

In  a  compound  term,  if  *t-'  is  followed 
by  *^'  or  *x',  a  parenthesis  must  be 


used  to  avoid  a  double  meaning. 


Illustrations 

20  =  10  +  6  -f  4, 

20  -  10  -  6  -  4  =  0, 

are  equations. 

In  the  first, 

20,  left  member; 
10  +  6  +  4,  right  member. 

In  the  second, 

20-10-6-4,  left; 
0,  right  member. 

In  both, 

20,  10,  6,  4,  terms. 


The  simple  term,  10, 
equals  any  one  of  the 
compound  terms ^  (8  +  2), 
(12  -  2),  6  X  2,  or  30  -^  3. 


6  —  2  +  5,  as  three  sim- 
ple terms;  or  (6—2  +  5),  as 
one  compound  term. 


5x2,  30  ^  3  ; 

not 

(5  X  2),  (30  --  3). 

24  -^  6  X  2  would  equal 
(24  -  6)  X  2, 
or  24-r-(6x2). 


60 


INTEC.ERS  61 

A  parenthesis  indicates  that  what  it  embraces  is  to  be  regarded  a« 
a  single  quantity. 

In  simplifying  expressions,  compound  terms  must  first  be  't'educed 
to  simple  terms. 

Readj  explain^  and  reduce  to  simple  terms : 

1.  6 -J. 3,  3  X  2,  (6  +  2).  5.  (8  +  6)(8-6),  (8+6)^(8-6). 

2.  (6-2),  (8-^ 2)h- 2,  8^(2-5-2).   6.  (8+6)-r-(5H-2),  6x(8h-2). 

3.  12  ^(4  +  2),  12  -5-(4  -  2).        7.  (8  -  5  +  7)-5-  2,  (8  -  2)  x  3. 

4.  3  X  (84-6),  3  X  (8-6),  3(8-6).  a  24^(6^2),  24 -i-(6  x  2). 

Ex.  2.  (8  H-  2)  H-  2,  the  expression,  8  -r-  2,  divided  by  2  ;  it  means  divide 
the  quotient  of  8  and  2,  by  2  ;  its  value  is  2.  8  h-  (2  -!-  2),  8  divided  by  the 
expression,  2  -t-  2  ;  it  means  divide  8  by  the  quotient  of  2  -r-  2  ;  its  value  is  8. 

Ex.  4.  3(8  —  6),  3  times  the  expression,  8  —  C  ;  the  sipi  'x'  is  usually 
omitted  before  a  parenthesis  ;  3(8  —  6)  means  3  times  the  difference  between 
8  and  6  ;  its  value  is  6. 

Write  as  an  equation : 

9.  That  3  times  the  sum  of  8  and  6,  multiplied  by  the  differ- 
ence between  12  and  9,  is  equal  to  126. 

10.  That  50,  diminished  by  the  product  of  6  and  3,  is  equal  to 
64  divided  by  the  quotient  of  8  and  4. 

11.  That  the  difference  between  8  and  5,  increased  by  7,  is 
equal  to  the  product  of  3  and  8,  diminished  by  the  quotient  of  28 
and  2. 

Find  the  value  of: 

12.  9  +  16  +  8-6-f-2.  la  10  +  12-h(4-f-2)-16-i-2. 

13.  8  -  25  -H  5  +  4  X  2.  19.  10  -  12  -s-(4  -  2)  + 16  x  2. 

14.  7-f  6x5-8-*-4.  20.  9-8-i-2-|-(7-5-|-4)-«-2. 

15.  9 -(6 -2)4- 2(8 -5).  21.  10  -  2  x  3  +  6  x  6 -8-!-2. 

16.  2  4-(9-2)-i-(10-3).  22.  3(9 -6 +  5)- 2(8 -5 +7). 

17.  9 +(6  4- 2)- 2(8 -5).  2a   (8  4- 7 -5)(8  -  7  4-5)- 2. 

Ex.  12.  Ans.  8.  Reducing  compound  terms  to  simple  terms,  9  +  2  —  3 1 
uniting,  8.     Ex.  22.   Ana.  4.     To  simple  terras,  24  -  20 ;  uniting,  4. 


62  OPERATIONS  COMBINED 

ANALYSIS 

In  solving  a  problem  by  analysis,  there  are  three  steps : 

The  meanings  and  the  relations  of  the  given  and  the  required 
terms  must  be  discovered. 

Each  relation  must  be  expressed  by  an  equation  in  such  a  way 
that  the  required  term  sliall  form  a  single  member  and  shall  not  be 
subjected  to  addition^  subtraction^  multijilicaJtiony  or  division. 

The  operations  suggested  in  the  relations  inust  be  performed. 

Terms  and  relations 
In  roost  problems,  the  given  and  the  required  terms  may  be 
recognized  and  understood  at  once,  and  the  relations  may  be  ascer- 
tained from  experience  or  from  a  knowledge  of  general  truths. 

State  the  given  termSj  the  required  terms^  and  the  relations: 

24.  After  losing  5^  a  boy  had  4^  left.     How  much  had  he  at 

first? 

Given  terms-  amount  lost,  bf;  amoaot  left,  if.  Required  term: 
amount  at  first.     Relation :  amount  at  first  =  amount  lost  +  amount  left. 

25.  After  losing  a  certain  sum  a  boy  had  4^  left     If  he  had 

9^  at  first,  how  much  did  he  lose  ? 

Given  terms:  amount  left,  4f ;  amount  at  first,  9f.  Required  term: 
amount  lost     Relation :  amount  lost  =  amount  at  first  —  amount  left 

26.  At  4/  each,  how  much  will  2  apples  cost  ? 

Given  terms :  cost  1  apple,  4  f  ;  number  apples,  2.  Required  term :  cost 
2  apples.     Relation :  cost  2  apples  =  2  x  cost  1  apple. 

27.  If  2  apples  cost  8^,  how  much  will  1  apple  cost  ? 

Given  terms :  number  apples,  2  ;  cost  all  apples,  8  f.  Required  term : 
cost  1  apple.     Relation :  cost  1  apple  =  I  cost  all  apples. 

2a  At  4^  each,  how  many  apples  can  be  bought  for  8^  ? 

Given  terms :  cost  1  apple,  4  ^  ;  cost  all  apples,  8  f.  Required  term : 
number  apples.  Relation :  number  apples  =  number  times  cost  of  all  con- 
tains cost  of  1. 


INTEGERS  63 

In  some  problems,  there  are  more  relations  than  one. 
29.   State  the  relations.     If  6  apples  cost  12^,  how  much  will  o 
apples  cost  ? 

Belatiom :  cost  1  apple  =  J  of  12  >» ;  cost  5  apples  =  6  times  cost  1  apple. 

In  some  problems,  relations  must  be  ascertained  from  a  knowl- 
edge  of  how  objects  are  constructed,  or  from  a  knowledge  of  the 
sciences. 

3a  State  the  principles  of  construction,  the  required  term,  and 
the  relation.  How  many  minute  spaces  has  the  hour  hand  of  a 
watch  passed  since  5  o'clock  when  the  minute  hand  is  at  3  ? 

Principles :  there  are  60  minute  spaces  on  a  dial ;  the  minute  liand  passes 
60  spaces  while  the  hour  hand  passes  5  spaces ;  etc.  (liven  term :  minute 
hand  has  passed  15  minute  spaces  since  5.  Required  term:  number  minute 
spaces  passed  by  hour  hand  since  5.  Relation :  number  spaces  passed  by 
hour  hand  =  ^^  number  spaces  passed  by  minute  band. 

In  some  problems,  the  relations  must  be  ascertained  from  a 
knowledge  of  business  usage. 

The  gain  equals  the  selling  price  minus  tfie  cost;  the  loss  equals 
the  cost  minus  the  selling  price. 

A  person  (principal)  may  give  pay  (commission)  to  another 
(agent)  for  buying  or  selling  articles  for  him.     Then  : 

The  entire  cost  equals  the  buying  j^rice  plus  the  commission;  the 
proceeds  equal  the  selling  price  minus  the  commission. 

ZL  State  the  relation.  If  an  orange  sells  for  6^  at  a  gain  of 
2  ^,  how  much  is  the  cost  ? 

Relation :  cost  =  selling  price  -  gain. 

32.  State  the  relation.  If  an  agent  buys  an  article  for  $50 
and  charges  a  commission  of  $1^  how  much  does  it  cost  the 
principal  ? 

3a  State  the  relation.  If  an  agent  sells  an  article  for  $50  and 
charges  a  commission  of  $  1,  how  much  does  the  principal  receive  ? 


64  OPERATIONS  COMBINED 

Write  the  relations;  do  not  solve  the  problems. 

34.  In  jumping,  A  beats  B  by  2  feet.  If  A  jumps  10  feet,  how 
far  does  B  jump  ? 

35.  In  jumping,  A  beats  B  by  2  feet.  If  B  jumps  10  feet,  how 
far  does  A  jump? 

36.  If  3  men  can  do  a  piece  of  work  in  6  days,  in  how  many 
days  can  1  man  do  it  ? 

37.  If  1  man  can  do  a  piece  of  work  in  18  days,  in  how  many 
days  can  3  men  do  it  ? 

3a  With  his  present  force,  a  contractor  can  do  a  piece  of  work 
in  18  days.  By  what  number  must  he  multiply  his  force  to  finish 
the  contract  in  2  days  ? 

39.  If  it  takes  a  man  10  minutes  to  saw  a  log  into  3  pieces,  how 
long  will  it  take  him  to  saw  it  into  4  pieces  ? 

4a  When  the  hands  of  a  watch  are  opposite  to  each  other,  how 
many  minute  spaces  are  there  between  them  ? 

41.  Three  boys  bought  a  top  for  10^ ;  the  first  gave  2^,  and  the 
second,  4^.     How  much  did  the  third  give  ? 

42.  A  and  B  travel  in  the  same  direction,  A  at  the  rate  of  5 
miles  per  hour,  and  B  at  the  rate  of  7  miles  per  hour.  How  many 
miles  does  B  gain  in  9  hours  ? 

43.  Conditions  as  in  Ex.  42,  if  A  has  a  start  of  4  hours,  in  how 
many  hours  will  B  overtake  him  ? 

44.  If  they  start  from  the  same  place  and  at  the  same  time  and 
travel  in  opposite  directions,  in  how  many  hours  will  they  be  36 
miles  apart  ? 

45.  By  selling  a  watch  for  $90,  a  man  would  gain  $20;  at 
what  price  must  he  sell  it  to  gain  $25? 

46.  If  $  4.20  is  paid  for  3  days'  work,  how  much  will  be  paid 
for  10  days'  work  ? 


INTEGERS  65 

The  process  as  a  whole 

The  relation  will  often  suggest  a  better  method  than  the  set  forms  of 
analysis.    Seepages  S6,  55,  4^^  57. 

47.  If  3  apples  cost  6^,  how  much  will  12  apples  cost  ? 

Skt  Form.  —Since  3  apples  cost  6j^,  1  apple  will  cost  ^  of  6/*  or  2j* ; 
since  1  apple  costs  2  f,  12  apples  will  cost  12  times  2  f,  or  24^. 

Better  Form.  —The  cost  of  12  apples  is  4  times  the  cost  of  3  apples,  or 
4  times  6  f,  or  24  f. 

Good  judgment  should  be  used  in  selecting  the  forms. 

4a  What  is  the  profit  on  buying  6  cows  at  $26  each  and  sell- 
ing them  at  $  28  each  ? 

GrooD  Judgment.  — The  profit  on  1  cow  is  the  difference  between  $28  and 
$26,  or  $2  ;  the  profit  on  6  cows  is  6  times  $2,  or  ^  12. 

Poor  Judgment.  —  Since  1  cow  costs  $26,  6  cows  will  cost  6  times  $26,  or 
8 156 ;  since  1  cow  sells  for  $28,  6  cows  will  sell  for  6  times  $28,  or  $  168 ; 
the  profit  is  the  difference,  or  $  12. 

In  vfritten  work,  norite  what  each  term  represents. 

49.  Through  an  agent,  I  sell  8  horses  at  $58  each,  commission 
$  1  per  head,  and  buy  with  the  proceeds  cows  at  $  26  each,  com- 
mission $  1  per  head.     How  much  should  the  agent  remit  ? 

f  58,  sell.  p.  1  horse  f  26,  cost  1  cow 

i,  commission  1,  commission 

67 f  proceeds  1  horse  27,  entire  cost  1  cow 

S  27)  456  {16  f^ 

456,  entire  proceeds  24,  amount  to  remit 

Be  sure  to  prove  every  answer. 

5a  Prove  in  Ex.  49,  that  the  agent  should  remit  $  24. 

$27,  cost  1  cow 

J6  Since  the  entire  cost  of  tlie  cows 

1^,  entire  cost  cores  P^"^  ^^«  '^"^^""^  remitted  Js  the  pro- 

^  .  ^  .^^    ,  ceeds  from  the  sale  of  the  horses,  the 

_24y  amount  remitted  ^„3^^^  j^  ^^^ 

456,  amount  to  invest  in  cows 

AMER.    ARITH.  —  5 


66  OPERATIONS  COMBINED 

51.  In  an  election,  Mr.  Jones  received  3689  votes,  but  was  de- 
feated by  216  votes.     How  many  votes  did  his  opponent  receive  ? 

52.  In  an  election,  Mr.  Brown  received  3905  votes  and  defeated 
his  opponent  by  216  votes.  How  many  votes  did  his  opponent 
receive  ? 

53.  I  bought  a  horse  for  $  15,284,  paying  $  2684  cash  and  the 
balance  in  monthly  payments  of  $1575  each.  How  many 
monthly  payments  did  I  make? 

54.  A  farm  house  is  worth  $  2450 ;  the  farm  is  worth  12  times 
as  much,  less  $  600 ;  and  the  stock  is  worth  twice  as  much  as  the 
house.     How  much  are  the  house,  stock,  and  farm  worth  ? 

55.  In  still  water,  a  crew  can  row  10  miles  per  hour  j  the  cur- 
rent runs  2  miles  per  hour.  How  many  miles  can  they  row  down 
stream  in  1  hour  ?     State  the  relation. 

56.  How  many  miles  can  they  row  up  stream  in  1  hour  ?  State 
the  relation. 

57.  In  how  many  hours  can  they  row  48  miles  down  stream 
and  return  ?     State  the  relations. 

5a  A  crew  can  row  down  stream  12  miles  per  hour  and  up 
stream  8  miles  per  hour.  What  is  the  rate  of  the  current  ?  State 
the  relations. 

59.  In  how  many  hours  could  the  crew  in  example  58  row  30 
miles  in  still  water  ?     State  the  relations. 

60.  How  much  will  5  barrels  of  potatoes  cost  if  13  barrels 
of  apples  cost  $  39,  and  6  barrels  of  apples  cost  as  much  as  9 
barrels  of  potatoes  ?     State  the  relations. 

61.  If  5  horses  eat  14  bushels  of  oats  in  2  Aveeks,  how  long 
would  it  take  them  at  the  same  rate  to  eat  56  bushels  ? 

62.  E  owes  a  debt  of  $  365.  How  many  sheep  must  he  sell  at 
$  15,  commission  $  1  each,  to  discharge  the  debt  ?  How  much 
money  will  he  have  left  ? 

63.  A  man  owed  ^2896;  he  paid  $499  at  one  time,  and  all 
but  $  375  a  second  time.     How  much  did  he  pay  the  second  time  f 


INTKGKRS  67 

64.  If  37  horses  cost  $  1295,  how  much  will  48  horses  cost  ? 
State  the  relations. 

65.  How  much  will  126  barrels  of  beans  cost  if  9  barrels  cost 
$  22  ?     State  the  relation. 

66  If  an  orchard  is  sold  for  $  375  at  a  loss  of  $  28,  what  was 
the  cost  ?     State  the  relation. 

67.  If  a  stock  of  goods  costing  $4376  is  sold  at  a  gain  of 
$  1094,  what  is  the  selling  price  ?     State  the  relation. 

6a  B  sells  a  house  through  an  agent  and  receives  $  1975.  If 
the  agent's  commission  is  $  97,  what  is  the  selling  price  ? 

69.  If  a  hound  runs  78  rods  while  a  hare  runs  64  rods,  how 
far  will  the  hound  run  while  the  hare  runs  1856  rods?  State 
the  relations. 

70.  Suppose  a  body  falls  16  feet  the  first  second,  48  feet  the 
next,  80  feet  the  next,  and  so  on,  constantly  increasing,  how  far 
will  it  fall  in  5  seconds  ?     State  the  relations. 

71.  Three  men  can  do  a  piece  of  work  in  5  days.  In  what  time 
can  1  man  and  8  boys  do  it,  if  1  man  does  the  work  of  2  boys  ? 
State  the  relations. 

72.  Conditions  as  in  Ex.  71,  how  many  boys  would  be  required 
to  do  the  work  in  one  day  ?     State  the  relations. 

73.  A  has  7  loaves  of  bread ;  B,  5 ;  C,  none.  The  three  eat  all 
of  the  bread,  each  the  same  amount.  C  pays  to  A  and  B  12  ^. 
How  much  should  each  receive  ? 

74.  A  liveryman  makes  an  annual  profit  of  $125  from  each 
horse ;  his  income  each  year  is  $  2125 ;  his  horses  cost  $  87  per 
head.  How  much  did  he  pay  for  the  horses?  State  the  rela- 
tions. 


FACTORING 


TERMS 

8  =  4x2. 
If  both  multiplicand  and  multiplier  are  wanting,  this  becomes 

8  =  what  X  what? 
It  means,  "what  are  the  numbers  whose  product  is  8 ?" 


Factoring  is  the  process  of  finding  num- 
bers whose  product  is  given.  The  numbers 
required  sae  factors  or  measures;  the  prod- 
uct, a  multij)le. 

Every  integer  is  the  product  of  itself  and 
one. 

If  a  number  has  no  set  of  integral  factors 
besides  itself  and  one,  it  is  a  prime  number. 
If  it  has  another  set  besides  itself  and  one, 
it  is  a  composite  number. 

Numbers  are  prime  to  each  other  when 
their  greatest  common  factor  is  one. 

Numbers  are  severally  prime  when  each 
is  prime  to  each  of  the  others. 


What  are  the  factors 
of  30? 

Am.  S  and  75,  S  and 
iO,  5  and  6^  or  SO  and 
J;  SO  =  e  X  15,  SxlO, 
6  x6t  or  SO  X  J. 


7,  a  prime  number; 
30,  a  composite  number. 


4,  8,  9,  are  prime  to 
each  other. 

4,  9,  25,  are  severally 
prime. 


FROM  THE  COMBINATIONS 

If  the  combinations  in  multiplication  are  known,  the  factors  of 
all  numbers  less  than  100  may  be  called  rapidly.    See  pp.  36,  4^. 

State  sets  of  two  factors  for  : 

1.  99,  98,  96,  95,  94,  93,  92,  91,  90,  88,  87,  86,  85,  84,  82, 

81,  80,  78,  77,  76,  75,  74,  72,  70,   69,   68,   66,   65,   64,   63,  62, 

60,  58,  57,  56,  55,  54,  52,  61,  50,  49,  48,  46,  45,  44,  42,  40. 

Ex.  1.    Ans.  96  =  2  X  48,  3  X  32,  4  X  24,  6  X  16,   8  x  12 ;  .  .  .  . 


INTEGERS 


69 


BY  INSPECTION 

Whether  one  of  the  factors  of  a  iiuiiiImt  is  2,  3,  4,  5,  8,  9, 11,  or  a 
product  of  two  or  more  factors  severally  prime,  may  be  found 
by  the  following  principles.     See  p.  80. 


A  number  is  divisible  by  2,  when  the 
number  denoted  by  its  last  digit  is  di- 
visible by  2,  or  is  0. 

A  number  is  divisible  by  5,  when  the 
number  denoted  by  its  last  digit  is  di- 
visible by  5,  or  is  0. 

A  number  is  divisible  by  4,  when  the 
number  denoted  by  its  last  two  digits  is 
divisible  by  4. 

A  number  is  divisible  by  8,  when  the 
number  denoted  by  its  last  three  digits 
is  divisible  by  8. 

A  number  is  divisible  by  3,  when  the 
sum  of  its  digits  is  divisible  by  3. 

A  number  is  divisible  by  9,  when  the 
sum  of  its  digits  is  divisible  by  9. 

A  number  is  divisible  by  11,  when  the 
difference  between  the  sum  of  its  digits 
in  the  odd  places  and  the  sum  of  its 
digits  in  the  even  places,  is  divisible  by 
11,  or  is  0. 

A  number  is  divisible  by  the  product 
of  any  number  of  its  factors  which  are 
severally  ])rinie  to  each  other. 


Illustrations 

27725  is  divisible  by  2, 
because  6  is  divisible  by  2. 


28725  is  divisible  by  5, 
because  5  is  divisible  by  5. 


7.1112  is  divisible  by  4, 
because  12  is  divisible  by  4. 


21816  is  divisible  by  8, 
because  816  is  divisible  by 
8. 

27810  is  divisible  by  3, 
because  18,  the  sum  of  its 
digits,  is  divisible  by  3. 

27810  is  divisible  by  9, 
because  18,  the  sum  of  its 
digits,  is  divisible  by  9. 

1639  is  divisible  by  11, 
because  11,  the  difference 
between  16,  the  sum  of  its 
digits  in  the  odd  places,  and 
4,  the  sum  of  ita  digits  iu 
the  even  places,  is  divisible 
by  11. 

27720  is  divisible  by  7. 8, 
9,  and  Ls  therefore  divisible 
by  7  X  *  X  5. 


70 


FACTORING 


Of  (he.  numbers  2^  S,  J^  5,  8y  P,  //,  use  those  tvhich  are  severalbj 
prhnej  and  form  comhinalions : 


2.  Of  2  and  one  other, 
a  Of  3  and  one  other. 

4.  Of  4  and  one  other. 

5.  Of  5  and  one  other. 

6.  Of  8  and  one  other. 
Ex.  2.  2  X  3,  2  X  6,  2  X  0, 2  X  11. 

StaJte  why  27720  is  divUible  : 
12.  By  2;  3;  4;  6. 
la   By  7;  8;  9;  11. 

14.  By  2  X  3;  2x5;  2x9. 

15.  By  2x11;  3x4;  3x5. 
Ex.  13.  By  7,  by  trial. 

By  inspection  tell  why . 
2a  2448  is  divisible  by  72. 

21.  6930  is  divisible  by  55. 

22.  4788  is  divisible  by  63. 

23.  8184  is  divisible  by  88. 

Ex.  20.  2448  is  divisible  by  8  and  9, 
fore  by  8  X  9,  or  72. 

Find  all  the  factors  of: 

28.  12540,  less  than  100. 

29.  27720,  less  than  100. 

Ex.  28.  2,  3,  4,  5,  11 ;  6,  10,  22,  12, 

Note.  —  It  is  best  to  test  for  2,  3,  4,  5, 
factors  which  are  severally  prime. 


7.  Of  9  and  one  other. 

a  Of  2,  3,  and  one  other. 

9.  Of  2,  5,  and  one  other. 
10.  Of  5,  8,  and  one  other. 
U.  Of  5,  8,  9,  and  one  other. 
Ex.  a  2  X  3  X  5,  2  X  3  X  11. 


la  By  3x8;  3x11;  4x5. 
17.  By  4x9;  4x11;  6x9. 

la  By  5x8x9;  8x9  xll. 
19.  By  3  X  7 ;  5x8x9x11. 
Ex.  la  3  and  8  are  severally  prime. 


24.  20934  is  divisible  by  18. 

25.  14630  is  divisible  by  77. 
2a  30144  is  divisible  by  24. 
27.  98000  is  divisible  by  35. 
which  are  severally  prime,  and  there- 


30.  17622,  less  than  100. 

31.  26585,  less  than  100. 
16,  33,  20,  44,  55 ;  30,  66 

7,  8,  9, 11 ;  and  then  to  combine  those 


INTEGERS  71 

Finding  prime  numbers 

Whether  an  integer  is  a  prime  number  is  found  by  trial.  In 
the  trial,  it  is  necessary  actually  to  divide  only  by  7  and  by  prime 
numbers  greater  than  11;  divisibility  by  2,  3,  5,  11,  and  by  all 
composite  numbers  may  be  tested  by  inspection. 

32.  Is  397  a  prime  number  ? 

Ans.  Yes.  397  is  not  divisible  by  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  IS,  14, 
16,  16,  77,  18,  /i?,  nor  20.  397 -=- 20  =  19 +.  It  is  not  divisible  by  a  number 
larger  than  20,  for  the  quotient  would  then  be  a  number  that  has  already 
been  tried. 

NoTK.  —In  this  example,  to  test  divisibility  by  7,  13,  17,  and  19,  the  division 
must  be  performed. 

33.  Make  a  list  of  all  the  prime  numbers  to  100.  How  many 
are  there  ? 

34.  Is  431  a  prime  number  ?    323?    131?    523?    601?    319? 

35.  Find  the  prime  number  next  greater  than  401. 

Expressing  by  factors 

36.  Express  2772  by  factors.         37.   Express  by  prime  factors. 

12)2772  2772  =  12  x  11  x  21. 

11^2 '91  ^"  prime  factors, 

^-^  2772  =  2x2x3x11x3x7 
'^^  =  22  X  32  X  11  X  7. 

2772  =  12  X  11  x21. 

NoTK.  —  A  number  written  over  a  factor  shows  how  many  times  the  factor  is 
used. 

Express  by  factors :  Express  by  prime  factors  : 

3a   72;   630;   2808.  42.   120;   220;   5616. 

39.   56;   836;    7425.  43.   108;    144;    1445. 

4a  48;   840;   1232.  44.   156;   160;   7392. 

41.  45;   945;   5929.  45.   135;   288;   3025. 

Note.— To  express  by  prime  factors,  it  is  best  to  re<!uce  to  factors  as  most 
convenient,  and  then  to  reduce  all  composite  to  prime  factors.  Do  not  form  the 
habit  of  always  dividing  by  2  and  by  3  ;  try  12,  U,  9,  8,  7,  6,  4,  3,  2,  in  order. 


72 


FACTOKING 
THE  FOUR  OPERATIONS 


It  is  often  convenient  to  atld  or  subtract  niunbers  each  of  which 
is  expressed  hy  two  facrtors.  We  add  or  subtract  the  factors  not 
common  and  retain  the  common  factor. 


46.   Add  16  X  7  and  22  x  7 

16  y.  7 
22x7 


47.   From  38  x  7  subtract  16  x  7 

S8y.7 
16x7 


38x7 


X  7 


16  X  7  is  16  sevens;  22  x  7  is  22  38  x  7  is  38  sevens;  16  x  7  is  16 
sevens.  Tlie  sum  is  38  sevens^  or  sevens.  Tlie  difference  is  22  sevens, 
38  X  7.  or  22  x  7. 


It  is  often  convenient  to  multiply  or  divide  numbers  expressed 
by  factors.  We  use  each  factor  of  the  multiplier  or  the  divisor, 
together  with  only  one  factor  of  the  other  term. 


4a   ^lultiply  2  X  3  by  5. 

2x3  3x2 

6  6 


10x3 


15x2 


2  X  3  is  2  threes,  or  3  iioos ;  5  times 
2  threes  is  10  threes,  or  10  x  3. 
5x3  twos  is  15  tiDOS,  or  15  x  2. 

50.   Divide  4  x  6  by  2. 

2 

^^  =  2x6 

4  sixes  -J-  2  =  2  sixes ;  or  6  fours 
-5-2  =  3  fours. 


49.   Multiply  2x3x5  by  4>^ 6. 

2x    3x    5 
4x6 


2  X  12x30 

(2  X  3  X  5)  X  (4  X  6) 
=  2x(3x4)x(5x6),  or 
2  X  12  X  30. 

51.   Divide  2x12x30  by  4x6. 


3       5 
Jg  X  j2  X  g0 
i4x(5 


2x3x5 


12  -=-  4  =  3  ;  30  -T-  6  =  5  ;  the  result 
is  2  X  3  X  6. 


NoTB.  —  This  method  is  culled  cancellation.    See  pp.  90,  92. 


INTEGERS  73 

Written  work 
Add :  Subtract  : 

52.  89  X  15,  16  X  15,  13  x  15.       54.  75  x  15  from  89  x  15. 

53.  7x12,    8  X  12,    9  X  12.       55.  16  x  17  from  73  x  17. 

Multiply :  Divide  : 

56.  2  X  3  X  4  by  6;  by  8.     5a  12  x  15  x  21  by  3;  by  6. 

57.  8  X  9  X  7  by  3;  by  7.     59.  21  x  35  x  42  by  7;  by  3. 

Expi'ess  the  quotients  by  factors  : 
ea  (18x9xl6)-i-(2x3x3).    65.  (48x96x98)-?- (16x24x49). 

61.  (16  X  18  X  12)--(4x9x2).    66.  (36 x 24 x 90) h- (18x18x12). 

62.  (72x24x36)H-(18x3x3).    67.  (18  x  12  x  30)-5-(5  x  6  x  3). 

63.  (75xl8x21)-^(35x9x5).    6a  (94 x 78 x 65) -- (47 x  13 x 39). 

64.  (56x49x63)-h(14x7x9).    69.  (57 x 91  x 77) -h (19x13x11). 

Mental  work 
Divide,  declaring  the  results  by  factors  : 
7a  85  X  6  X  7,  by  17.  7a  17  x  19  x  18,  by  34. 

71.  95  X  8  X  3,  by  24.  79.  23  x  12  x  15,  by  69. 

72.  72  X  8  X  7,  by  56.  80.  26  x  27  x  28,  by  63. 

73.  90  X  3  X  7,  by  54.  at  96  x  35  x  17,  by  48. 

74.  75  X  4  X  8,  by  50.  82.  80  x    9  x  11,  by  88. 

75.  18  x  8  X  4,  by  36.  83.  62  x    8  x  63,  by  93. 

76.  16  X  7  X  8,  by  64.  84.  22  x    3  x  33,  by  66. 

77.  19  X  9  X  4,  by  38.  85.  18  x    4  x  62,  by  93. 

Ex.73.   Ans.  6x7.    The  factors  of  64  are  18  and  3;    90+18  =  6; 
3  +  .3  =  1. 

NoTK.  — For  practical  illustrations  of  Exs.  70  to  85,  see  Notes,  pp.  88,  89. 


74  FACTORING 

GREATEST  COMMON  DIVISOR 

The  greatest  common  divisor  of  numbers  less  than  100  may  be 
known  from  the  combinations.     See  pp.  36  and  40. 

Findf  from  the  combinationsy  the  O.  C.  D.  of: 

86.  40,  96.  89.   30,  48.  92.  48,  80. 

87.  24,  66.  9a  64,  72.  9a  46,  94. 
8a  35,  91.                 91.   66,  60.  94.  49,  77. 

Kx.  86.   8.     40  +  8  =  5 ;  96  -^  8  =  12  ;  5  and  12  are  prime  to  each  other. 

Smaller  numbers  may  be  found  which  have  the  same  G.  C.  D. 

I.    Hie  G.  C.  D.  of  two  numbers  is  the  O.  C.  D  of  the  smaller,  and 
of  the  remainder  found  by  dividing  the  greater  by  the  smaller. 

95.  In  Exs.  86  to  94,  find  smaller  numbers  with  same  G.  C.  D. 

Ex.  86.   Ans.   40,  16.    Since  06  =  40  +  40  +  16,  the  G.  C.  D.  of  40  and  96 
is  the  G.  C.  D.  of  40,  and  40  +  40  +  16,  or  of  40  and  16. 

96.  Making  use  of  the  first  expedient,  solve  examples  86  to  94 

Ex.  86.  Ans.  8.    The  G.  C.  D.  of  40  and  96  is  the  G.  C.  D.  of  40  and 
16,  or  8. 

n.    One  of  several  numbers  may  be  divided  by  a  fa/dor  prime  to 
any  other,  without  affecting  the  G.  C.  D. 

97.  In  Exs.  86  to  94,  find  smaller  numbers  with  same  G.  C.  D. 

Ex.  86.   Ans.  8,  96.     Since  40  contains  6,  which  is  prime  to  96,  5  may 
be  canceled  from  40  without  affecting  the  G.  C.  I). 

98   Making  use  of  the  second  expedient,  solve  examples  86  to  94. 
Ex.  86.   Ans.  8.     The  G.  C.  D.  of  40  and  96  is  the  G.  C.  D.  of  8  and  96, 


or  8. 


Find  the  G.  C.  D  of: 

99.  36,  40,  48,  72.  102.  55,  77,  88,  99. 

100.  49,  5(S,  70,  77.  103.  45,  75,  90,  60. 

101.  60,  84,  96,  72.  104.  42,  56,  70,  77. 


INTEGERS  76 

III.  The  G.  C.  D.  of  tv3o  or  more  numbers  is  the  product  of  all 
the  common  factors  that  may  be  used  as  successive  divisors  until  the 
quotients  are  prime  to  each  other. 

Find  the  O.  C.  D  of: 

105.  288,  432,  720.  109.  945,  1260,  2625. 

106.  495,  660,  990.  110.  1680,  4200,  5040. 

107.  475,  855,  760.  111.  1875,  3750,  5000. 

loa  1728,  1296,  1872.  112.  2850,  3800,  4750. 

Ex.  105.  By  expedient  III. 

12  I  288    432    720  Since   12  and  12  are  component 

-^  — -^. ^ ^  factors  of  the  G.  C.  I).,  and  the  quo- 

2        3        5  or  144,  is  the  G.  C.  D. 

Note.  —  In  finding  common  factors  by  inspection,  it  is  best  to  try  12, 11,  9,  8, 
7, 5,  4,  3,  2,  in  order. 

If  no  common  factor  is  seen  by  inspection,  smaller  numbers 
should  be  found  which  have  the  same  G.  C.  D.     ^ee  p.  7 4. 

Find  the  G.  C.  D.  of: 

113.  153,  374.  117.  143,  1765,  2912. 

114.  625,  1728.  lia  495,  1452,  9317. 

115.  1144,  1365.  119.  1152,  1728,  3375. 

116.  1177,  2675.  120.  875,  448,  567. 
Ex.  113.   By  expedient  I. 

153)374(2 

^0^  The  G.  C.  D.  of  163  and  374  is  the 

68)153(2  G.  C.  D.  of  1 63  and  08,  or  the  G.  C.  D. 

136  of  68  and  17,  or  17. 

17 

Ex.  113.   By  expedient  IL 

153     374  The  G.  C.  D.  of  163  and  374  is  the 

51  G.C.  D.  of  51  and  374,  or  the  G.  C.  D. 

27  of  17  and  374,  or  17. 


76  FACTORING 

LEAST  COMMON  MULTIPLE 

Multiples  of  small  numbers  are  easily  found  from  the  combina- 
tions.    See  pp.  36,  4^. 

1.  Hie  least  common  multiple  of  two  or  mot'e  numbers  is  the 
product  of  all  their  prime  factors,  each  taken  the  greatest  number 
of  times  it  is  found  in  any  one  of  them. 

121.  Write  all  the  multiples  of  2,  to  36 ;  all  the  multiples  of  3, 
to  36. 

122.  Make  a  list  of  all  the  multiples  of  2,  less  than  36,  that 
exactly  contain  3;  of  all  the  multiples  of  3,  less  than  36,  that 
exactly  contain  2. 

12a  What  is  the  least  common  multiple  of  2  and  3  ?  Is  it 
exactly  contained  in  each  of  the  common  multiples  ?    Why  ? 

124.  Answer  examples  121  and  122,  with  reference  to  4  and  6, 
instead  of  2  and  3. 

125.  What  is  the  least  common  multiple  of  4  and  6?  Is  it 
exactly  contained  in  each  of  the  common  multiples  ?    Why  ? 

126.  Why  is  2  X  3  the  least  common  multiple  of  2  and  3,  while 

4  X  6  is  not  the  least  common  multiple  of  4  and  6  ? 

Ans.  2x3  is  the  L.  C.  M.  of  2  and  3  because  2  and  3  are  prime  to  each 
other ;  4  X  G  is  not  the  L.  C.  M.  of  4  and  6  because  4  and  6  are  not  prime. 

Find,  from  the  combinations,  the  L.  C.  M.  of: 

127.  10,  18.  131.  16,  20.  135.  21,  14. 
12a  14,  12.                    132.   12,  16.  136.   12,  18. 

129.  10,  12.  133.   24,  16.  137.   24,  32. 

130.  10,  14.  134.  25,  20.  13a  27,  18. 

139.  Taking  the  product  of  prime  factors,  solve  Exs.  127  to  138. 

Ex.  136.  The  L.  C.  M.  must  contain  12,  or  2  x  2  x  3, 

12  —  Q  ^  S  \f  «  *"^  ^^'^  retain  these  factors.     The  L.  C.  M. 

must  contain  18,  or  2  x  3  x  3.     We  already 

18  •=^  2  X  3  y.  S  l^ave  2  and  one  of  the  3's  ;  we  retain  the 

8x2 X3x 3=36,  L.  C.  M.       other.    Then 2  x  2  x  3  x  3  is  the  L.  C.  M. 


INTEGERS  77 

The  product  of  the  prime  factors  may  be  found  more  easily  by 
the  following  principles : 

II.  To  find  the  L.  C.  M  of  two  numbers^  divide  one  of  them  by 
their  O.  C.  D  and  midtiply  the  quotient  by  the  other. 

III.  To  find  the  L.  C.  M.  of  more  than  two  nu7nber.%  find  the 
L.  CM  of  two  of  them,  then  of  the  result  and  a  third,  and  so  on. 

IV.  If  one  of  the  numbers  exactly  contains  another,  the  smaJUer 
may  be  neglected. 

140.  In  Exs.  127  to  138,  find  the  L.C.M.  by  principle  II. 

Ex.  136. 

12,  18  As  before,  the  L.  C.  M.  must  contain 

12  y.  S  =i  S6  L  C  M  12,  or  2  x  2  x  3.     Instead  of  retaining 

'     '     *      '  these  factors,  we  retain  12  itself.    The 

The  G.C.  D.  of  12  and  18  is  6 ;      L.  C.  M.  must  contain  18,  or  2  x  3  x  3. 

18  -4-  6  is  3  ;  12  X  3  is  30,  L.  C.  M.       In  12,  we  already  have  2  once  and  3 

once,  or  6,  the  G.  C.  D.  of   12  and  18, 
and  we  simply  retain  the  other  3,  or  18  -f-  G.  C.  D. 

NoTK.  — By  retaining  12,  instead  of  2  x  2  X  3,  we  are  saved  the  effort  of  first 
separating  12  into  its  factors,  and  later  of  multiplying  them  together. 

Find  mentally  the  L.  C,  M.  of: 

141.  60,  72.  145.   16,  20.  149.  21,  45. 

142.  25,  30.  146.  20,  45.  150.  16,  30. 

143.  24,  27.  147.  24,  36.  151.  24,  30. 

144.  49,  63.  14a  42,  56.  152.  60,  75. 

Ex.  141.   00  X  6,  or  360.    The  G.  C.  D.  of  60  and  72  is  12 ;  72  h-  12  =  6  ; 
60  X  6  =  360,  L.  C.  M. 

By  these  principles,  find  mentally  the  L.  C.  M.  of: 

153.  2,  3,  4,  5,  6,  7.  155.  10,  12,  15,  6,  5,  4. 

154.  5,  8,  9,  6,  12,  10.  156.   18,  8,  9,  36,  24,  Vz. 

Ex.  153.    Ans.  420.    We  neglect  2  and  3  because  they  are  contained  in  A 
The  L.  C.  M.  of  4  and  6  is  20 ;  of  20  and  0,  60 ;  of  60  and  7,  420. 


78  FACTORING 

Another  method  of  finding  the  L.  C.  M.  is  to  divide  by  prime 
divisors  that  are  common  to  any  two  of  the  numbers. 

157.  Find  the  L.  C.  M.  of  63, 108,  28,  42. 

Since  3  is  contained  in  63  and  108,  3  is 
used  once.  Since  3  is  contained  in  21  and 
36,  3  is  used  a  second  time.  Since  2  is  con- 
tained in  12  and  28,  2  is  used  once.  Since  2 
is  contained  in  6  and  14,  2  is  used  a  seanul 
time.  Since  7  is  contained  in  7  and  7,  7  is 
i         S       1       1  used  once.     Since  3  is  among  the  quotients, 

3^X2^X7=  756,  L.  CM.      3  is  used  a  third  time. 

If  no  two  numbers  appear  to  have  a  common  factor,  it  is  neces- 
sary to  find  their  G.  C.  D. 

15a  Find  the  L.C.M.  of  692,  703, 171. 


s 

63 

108 

28 

^ 

s 

21 

36 

28 

U 

2 

7 

12 

28 

U 

2 

7 

6 

u 

7 

7 

7 

3 

7 

7 

37 
19 


703    171 


16       19     171  Th«  G.  C.  D.  of  592  and  703  is 

-7^ Z r  37.     The  G.  C.  D.  of  19  and  171  is 

1^         ^         ^  19. 

37xl9xl6x  9=101,232,  L.  C.  M. 

159.  Solve  Ex.  157  again.  Divide  by  4;  then  select  such 
divisors  as  you  please.     Why  is  the  answer,  1512,  incorrect? 

Ans.  Because  the  greatest  number  of  times  2  is  used  in  any  number  is 
twice^  viz.,  in  108  and  28.  Dividing  by  4,  took  2  twice  out  of  108  and  28,  but 
left  2  once  in  42,  making  three  times  that  it  appears  by  this  process. 

Note.  —  It  is  ansafe  to  divide  by  other  than  prime  numbers. 

Find  the  L.C.M.  of: 

160.  30,  32,  45,  48,  64,  75.  16a  169,  221,  204. 

161.  125,  245,  147,  225.  164.  266,  285,  209. 

162.  36,  72,  128,  126,  243.  165.  708,  1062,  1475. 


INTEGERS  79 

Casting  out  9*8 

Addition,   subtraction,    multiplication,   and   division   may  be 
proved  by  casting  out  U's. 

In  addition^  the  sum  of  the  excesses  of  9*8  in  tlie  several  addends^ 
slioidd  equal  the  excess  ofO^s  in  the  sum. 

5078       8  ^Ve  add  the  digits  of  the  first  addend,  5,  11,  18, 

A93A       s  ^^ '  ^**^"  ^^^  digits  of  the  sum,  26,  and  write  8. 

■*      ■*  This  is  the  excess  of  the  9's,  for  5678  h-  9  gives  8 

^^^      _^  for  a  remainder.     We  proceed  in  the  same  way 

1A£39        1  ^'^**  ^^'^  °^^^^  addends  :  4,  13,  16,  20  ;  2  :  3,  9,  11, 

"^^  18 ;  9 ;  0  (a  sum  of  9,  or  any  multiple  of  9,  we 

count  as  0).     We  proceed  in  the  same  way  with  the  excesses,  8,  10  ;  1. 
We  proceed  in  the  same  way  with  the  sum,  1,  6,  7,  10,  19 ;  10  ;  1. 
The  sum  of  the  excesses  of  9's  in  the  several  addends  is  1  ;  the  excess  of 
9*8  in  the  sum  is  I ;  the  answer  is  probably  correct. 

In  subtraction,  the  excess  of  9*s  in  the  minuend,  should  equal  the 
excess  of  9^s  in  the  subtraJiend  and  the  remainder. 

'^^^^       ^  Minuend,  7,  13,  21,  24;  6. 

S997      J^  Subtrahend,  3,  12,  21,  28  ;  10;  1 :  remainder, 

~J^      7  ^'  ^'  ^^'  23;  b:  their  sum,  6. 

In  multiplication,  the  excess  of9*s  in  the  product  of  the  excesses  of 
the  factors,  should  equal  the  excess  in  the  answer. 

^^^       ^  Multiplicand,  9,  16,  23 ;  6 :  multiplier,  4,  10, 

468       0  18  ;  9  ;  0 :  their  product,  0. 


463,0^4       0  ^^«^"<^^'  ^'  ®'  ^2,  14,  18 ;  9  ;  0. 

In  division,  the  excess  of  9'8  in  the  product  of  the  excesses  of  the 
quotient  and  divisor,  plus  the  excess  in  the  remainder,  should  equal 
the  excess  in  the  dividend. 

488)472,870968'^  Quotient,  9,  15,  23;  5:  divisor,  4,  12,  20;  2: 

their  product,  10 ;  1  :  the  remainder,  4,  12,  19 ; 
,  ^  10  ;  1  ;  their  sum,  2. 

—  -  Dividend,  4,  11,  13,  21,  28,  29 ;  11 ;  2. 

iO  » 


80  FACTORING 

RELATIONS 

166.  When  13,825,  or  13,820  -f  5,  is  divided  by  2,  why  is  the 

remainder  the  same  as  when  its  last  digit  is  divided  by  2  ?     Give 

the  rule  for  the  divisibility  of  a  number  by  2. 

The  first  part,  13,820,  or  1382  tens,  is  divisible  by  2  because  ten  is  divisible 
by  2.  Since  the  first  part  is  divisible  by  2,  the  divisibility  of  the  number 
depends  upon  the  last  digit. 

167.  When  13,825,  or  13,800  +  25,  is  divided  by  4,  why  is  the 
remainder  the  same  as  when  the  number  denoted  by  its  last  two 
digits  is  divided  by  4?     Give  the  rule. 

16a  When  13,825,  or  13,000  +  825,  is  divided  by  8,  why  is  the 
remainder  the  same  as  when  the  number  denoted  by  its  last  three 
digits  is  divided  by  8  ?     Give  the  rule. 

169.  What  is  the  remainder  when  1  with  any  number  of  ciphers 
is  divided  by  9  ? 

170.  What  is  the  remainder  when  2,  3,  4,  5,  6,  7,  or  8,  times  1 
with  any  number  of  ciphers,  is  divided  by  9  ? 

171.  When  13,825,  or  10,000  +  3000  +  800  -f-  20  +  5,  is  divided 

by  9,  why  is  the  remainder  the  same  as  when  the  sum  of  its 

digits  is  divided  by  9  ? 

When  10,000  is  divided  by  9  the  remainder  is  1 ;  when  3000  is  divided  by 
9  the  remainder  is  3  ;  etc.     See  Ex.  170. 

172.  If  a  number  divided  by  9  gives  the  same  remainder  as 
the  sum  of  its  digits  divided  by  9,  what  is  the  rule  for  the  divisi- 
bility of  a  number  by  9  ? 

173.  Show  that  1  with  any  odd  number  of  ciphers  lacks  1  of 
being  a  multiple  of  11. 

174.  Show  that  1  with  any  even  number  of  ciphers  exceeds  by 
1  a  multiple  of  11. 

175.  How  much  does  2,  3,  4,  5,  6,  7,  8,  9,  times  1  with  any  odd 
number  of  ciphers,  lack  of  being  a  multiple  of  11  ? 

176.  How  much  does  2,  3,  4,  5,  6,  7,  8,  9,  times  1  with  any  even 
number  of  ciphers,  exceed  a  multiple  of  11  ? 


INTEGERS  81 

177.  When  75,316,  or  (70,000 4-300 +6) +  (5000 +10),  is  divided 
by  11,  why  is  the  remainder  tlie  same  as  when  the  sura  of  the 
digits  in  the  odd  places  minus  the  sum  of  the  digits  in  the  even 
places,  is  divided  by  11  ? 

70,000+300  +  0  exceeds  a  multiple  of  11  by  7  +  3  +  6  (Ex.  176);  5000+10 
lacks  6  +  1  of  being  a  multiple  of  11  (Ex.  175);  75,316  exceeds  a  multiple  of 
llby  (7  +  3  +  0)-(5  +  l). 

17a  If  a  number  divided  by  11,  gives  the  same  remainder  as 
the  difference  between  the  sum  of  its  digits  in  the  odd  places  and 
the  sum  of  its  digits  in  the  even  places,  divided  by  11,  what  "is  the 
rule  for  the  divisibility  of  a  number  by  11  ? 

179.  A  contractor  is  to  build  houses  24,  36,  48,  and  60  feet 
long,  and  16,  32,  32,  and  48  feet  wide.  What  length  of  clapboard 
can  be  used  most  conveniently  for  the  sides?  for  the  ends? 
State  the  relations. 

Relation :  number  of  feet  in  length  of  clapboard  for  the  sides = the  largest 
number  that  is  exactly  contained  in  24,  36,  48,  and  00,  or  their  G.C.D.  .  .  . 

180.  A  lady  wishes  to  buy  a  piece  of  cloth  which  she  can  cut 
without  waste  into  an  exact  number  of  pieces  either  3,  4,  or  5 
yards  long,  as  she  may  decide  later.  What  is  the  smallest  num- 
ber of  yards  the  piece  can  contain  ? 

181.  A  real  estate  agent  wishes  to  divide  3  pieces  of  land  325, 
675,  and  950  feet  wide,  into  town  lots  of  equal  width.  What  is 
the  largest  possible  width  for  each  lot  ? 

182.  Ropes  48,  52,  and  56  feet  long  are  to  be  cut  into  the  long- 
est possible  equal  lengths.     How  long  must  each  piece  be  ? 

183.  A,  B,  and  C  start  together  around  a  circular  track.  A 
goes  once  around  in  6  minutes ;  B,  in  8  minutes ;  C,  in  9  minutes. 
AVhat  is  the  least  number  of  minutes  before  they  will  be  together 
again  at  the  starting  point  ?    State  terms  and  relations. 

(riven  terms :  number  of  minutes  passed  when  A  is  at  the  starting  point 
ill  6,  12,  18,  24,  30,  36,  42,  48,  54,  00,  66,  72,  ...  ;  B,  8,  16,  24,  32,  40,  48, 
66,  64,  72,  .  .  .  ;  C,  9,  18,  27,  36,  45,  54,  0.3,  72.  .  .  . 

Relation :  number  of  minutes  before  they  are  again  together  =  least  num- 
ber that  will  exactly  contain  6,  8,  and  9,  or  their  L.  C.  M. 

▲  MBR.    AHITH. —  6 


82  FACTORING 

184.  After  how  many  minutes  will  A  and  B  first  be  together  at 
the  starting  point  ?     A  and  C  ?     B  and  C  ? 

las.   D,  E,  and  F  start  together  around*  a  circular  track  5280 

feet  in  length.     D  rides  1760  feet  per  minute ;  E,  1320 ;  and  F, 

1056.     How  many  times  must  D  ride  around  the  track  before 

they  are  all  together  again  at  the  starting  point?    State  the 

relations. 

Relations:  number  of  minutes  D  goes  once  around  =  5280  -=-1760; 
number  minutes  B  =  5280  ^  1320  ;  number  minutes  C  =  5280  h-  1056.  The 
minute's  when  they  are  first  together  =  the  L.  C.  M.  of  the  minutes  each 
makes  the  circuit ;  number  times  A  goes  around  =  L.  C.  M.  -;-  number  min- 
utes A  makes  the  circuit. 

186.  By  counting  eggs,  4,  6,  or  10  at  a  time,  a  farmer  had  none 
left  over  in  each  case.  "What  is  Che  least  number  he  could  have 
had  ?    State  the  relations. 

187.  By  counting  eggs  4,  6,  or  10  at  a  time,  a  farmer  had  3  left 
over  in  each  case.  What  is  the  least  number  he  could  have  had  ? 
State  the  relation. 

18a  By  counting  eggs  4,  6,  or  10  at  a  time,  a  farmer  had  3  eggs 
left  over  in  each  case;  counting  11  at  a  time,  he  had  none  left. 
Wliat  is  the  least  number  he  could  have  had  ?    State  the  relations. 

Relations :  ppssible  numbers  =  3  -f  common  multiples  of  4,  6,  and  10  ;  the 
least  number  =  the  least  of  these  results  divisible  by  11. 

SohUion :  the  L.  C.  M.  of  4,  6,  and  10  is  60 ;  63,  123,  183,  243,  303, 
363,  .  .  .  are  numbers  representing  in  order  3  +  common  multiples  of  4,  6, 
and  10  ;  303  is  the  least  of  these  which  contains  11. 


COMMON   FRACTIONS 


FIRST  CONCEPTION 


AN  EXPRESSION  OF  DIVISION 


Division  may  be  expressed  by 
writing  the  dividend  above,  and 
the  divisor  below,  a  horizontal  line. 
Such  an  expression  is  a  common 
froA'tion;  the  dividend  is  the  nu- 
merator;  the  divisor,  the  denomi- 
nator. 

The  numerator,  or  the  denomina- 
tor, or  both,  may  contain  fractions ; 
such  an  expression  is  a  complex 
fraction. 

We  sometimes  speak  of  a  frac- 
tion of  a  fraction;  a  compound 
fraction. 

An  integer  plus  a  fraction  is  a 
mixed  number.  The  plus  sign  is 
usually  omitted. 

State  the  terms,  and  the  meaning 

of  the  fractions  :  } ;  5« 
i 


Illdstrations 

^,  common  fraction, 
5 

4y  numerator. 

5y  denominator. 

Read,  4-^5. 


S    '^   6 


8   complex 
"s   fractions. 
9 

a  compound 
fraction. 


5  -f-  ;,  or  5 


t    a  mix4>d 
5'  number. 


I ;  3  is  the  numerator  ;  4,  the 
denominator  ;  it  means  3  -r-  4. 

I  is  the  numerator ;  |,  the  de- 
nominator ;  it  means  f  -^  }. 

NoTR.  —The  pupil  should  read  the  first  part  of  p.  45,  the  whole  of  p.  49,  and 
the  explanation  of  Ex.  55  ou  p.  51. 

88 


84 


SECOND  CONCEPTION 


A  unit  may  be  divided  into 
two  or  more  equal  i)arts,  and 
one  or  more  of  these  parts  may 
be  considered. 

The  number  showing  into 
how  many  parts  the  unit  is 
divided,  is  written  below  a  hor- 
izontal line,  and  is  the  denomi- 
nator. 

The  number  showing  how 
many  parts  are  considered,  is 
written  above  the  line,  and  is 
the  numerator. 

The  whole  expression  is  a 
CO m mon  frart ion. 

According  to  this  conception, 
is  I  a  fraction?  No.  It  is 
called  an  improper  fraction,  i.e., 
not  properly  a  fraction. 

According  to  this  conception, 

IS    1^   a  fraction?     No. 

t 
called  a  complex  fraction. 


EQUAL  PARTS  OF  A  UNIT 

Illustrations 


It   is 


T — r 


AB  is  divided  into  8  equal  parts ; 
AC  contains  5  of  them  ;  AC  =  6 
eighths  of  AB ;  expressed,  AC  =  \ 
of  AB. 

g 

-,  common  fraction. 

o 

8t  denominator. 

5,  numerator. 

It  means  tliat  a  unit  is  divided 
into  8  equal  parts,  and  that  5  of 
tliese  parts  are  considered. 

\,  read,  one  half;  f,  read,  three 
quarters,  or  three  fourths. 

It  is  impossible  to  divide  a  unit 
into  5  equal  parts  and  then  con- 
sider 8  of  them. 


It  is  impossible  to  divide  a  unit 
into  I  equal  parts. 


Define  by  ea^h  conception : 

1.  A  common  fraction. 

2.  The  numerator. 

3.  The  denominator. 


4.  A  complex  fraction. 

5.  A  compound  fraction. 

6.  A  mixed  number. 


Note.  —  Sometimes  deductions  are  made  from  the  first  conception ;  and  some- 
times, from  the  second.  For  illustTations  of  the  first,  see  pp.  87  and  llO;  of  the 
second,  pp.  lOl^  102. 


COMMON   FRACTIONS 


85 


CHANGE  OF  FORM 
To  lower  terms 


Dividing  both  numerator  and 
denominator  by  the  same  num- 
ber does  not  change  the  value  of 
afrojction. 

By  this  principle,  we  reduce 
fractions  to  their  simplest  forms. 

Which  fraction  is  the  more 
readily  comprehended,  f  f  or  J  ? 
Why? 

7.  Reduce  ^fj  to  lowest 
terms. 

216     18     3 
a  Reduce    f^fj   to    lowest 


B 


I    '     I     '     I     '     I 

AB  =  i,  orf,  of  AC; 

••.*  =  ?. 
I  is  obtained  from  ^  by  dividing 
both  terms  by  2. 


I,  because  the  terms  are  smaller. 


By  inspection,  we  find  common 
factors.  Dividing  both  terms  by  12, 
m  =  H ;  by  6,  H  =  f . 


r-ry       in                                 When  no  ( 
5ob7       19                                hy  inspprtior 

6789     28                         here  it  is  293 

common  factor  is  found 
,  we  find  the  G.  C.  D. ; 

Reduce  to  lowest  terms : 

9-  li;  A-                   1*  Mi  «• 

17.  a 

«• 

laA;  M-                   i*-+li«- 

la  H 

«• 

11-  H;  H-                   "•  Mi  W 

19.  a 

H- 

"•   tti  ii                            16-   Hi  H- 

20.  a 

H 

Reduce  to  lotoest  terms : 

^  ttHi  /A^-          24.  im;  m 

27. 

tmi;   «M. 

»•  AV\,i  H«-           aa-  Wi!  Hi- 

2a 

TTtt»    AWr* 

^-    Ht*»    llli*                   *^    TtTTJ   Ht' 

29. 

AWriHi 

Itt- 

86 


CHANGE   OF    FORM 
To  higher  terms 


Multiplying  both  numerator 
and  denominator  by  the  same 
number  does  not  change  the 
value  of  a  fraction.  • 

By  this  principle,  we  prepare 


I  '  I  '  I  "n^ 

AB  =  i,  or  I,  of  AC ; 


I  is  obtained  from  J  by  multiply- 
fractions  for  addition  and  sub-     ing  both  terms  by  2. 
traction. 

aa  Change  }  to  12th8. 


S     12 
Change: 

31.  jtol2th8. 

32.  |to40ths. 
3a  ^to35th8. 
34.  {to48ths. 


To  make  the  denominator  12,  we 
must  multiply  3  by  4 ;  multiplying 
both  terms  by  4,  |  =  /j. 


35.  ■jS^to24th8. 

36.  T^to39ths. 

37.  -^toTTths. 
3a  '^to28ths. 


39.  |^tol35l8ts. 

40.  i^to6825th8. 

41.  ^to2856ths. 

42.  ^to2277ths. 


43.  Reduce  J,  f ,  f  to  equivalent  fractions  having  L.  C.  D. 

g       Q      2Q  The  L.  C.  D.  is   12  ;    12  -  3  =  4 ; 

— -,    — ,    — •  we  multiply  both  terms  of  f  by  4  ; 

12    12     12  12  ^  4  =  3  ;  we  multiply  both  terms 

of  J  by  3  ;  12  -=-  6  =  2  ;  we  multiply  both  terms  of  \  by  2. 


Reduce  to  equivalent  fractions  having  L.  C.  D : 

44.    I,     I,     |.  49.    I,  H,  A-  54.  f,  if,  IH 

55-  T>   TT?  TTS"' 

56-  l>  «.  Mi 

57- 1,  ^,  m 

5®-  'Si  wky  T9Z 


*5.    i,      \,     \. 

so.  i,    %,  if. 

46.    \,     1,  A- 

51.    f ,     i,  A- 

*'•  f.  A.  A- 

52.  i,  A.  H- 

4a  \,  ^,  A- 

S3,  f  A,  A- 

€OMMON   FRACTIONS 


87 


To  whole  or  mixed  numbers 

A  fraction  is  an  ex})ression  of  divi- 
sion. It  means  that  the  numerator  is 
to  be  divided  by  tlie  denominator. 

By  this  principle,  we  change  to  a 
form  that  can  be  more  readily  com- 
prehended. 

59.  Reduce  fj  to  a  whole  or  mixed  number. 
13 


3'-' 

IS 


Or,  since  there  are  3  thirds 
in  1,  in  8  thirds  there  are  as 
many  Vs,  as  3  is  contained 
times  in  8,  or  2|. 


f^=45~13.    Performing  the 
operation,  we  obtain  o/y. 


Reduce  to  a  wliole  or  mixed  number : 
ea  V;  i^.  63.   tl;  fj. 

^  «;  M-        6*-  tt;  ff- 

62.   H;  H-  65.    ^;  ff. 


66.  i-til. 

67.  if-fp. 

6a  ^^^. 


69.  i^i. 

70.  IfJfL. 

71.  ^ifF. 


Whole  or  mixed  numbers  to  common  fractions 


A  mixed  number  is  an  integer  plus 
a  fraction.  If  the  integer  is  reduced 
to  an  equivalent  fraction  having  the 
denominator  of  the  fraction,  the  two 
paHs  may  be  united. 

By  this  principle,  we  prepare  frac- 
tions for  multiplication  and  division. 


2f  =  2  +  f 
=  ?  +  ! 


Or,  since  there  are  3  thirds 
in  1,  in  2  there  are  2  times  3 
thirds,  or  6  thirds;  with  2 
thirds,  8  thirds. 


72.  Reduce  4j\  to  an  improper  fraction. 

Reduce  to  an  improper  fraction  : 
7a  5J;  4f  75.  5H;  6W. 

74.   7i;8i.  76.  3H;^A- 


4A  =  HH-A  =  lf. 


77.  35t|;  U^' 

7a  87H;  86H. 


88  ADDITION 

ADDITION 

Before  fractions  can  he  addedj  they  must  be 
reduced  to  equivalent  fractions  having  a  com-  ~  =  A 

mon  denominator,  . 

Tfie  least  common  denominator  should  alusxys  4~lg 

be  found. 

79.  Find  the  sum  of  }  and  |. 

UEOUN 

,  I  =  I'j ;  I  =  t'i  ;  the  sum  of 

5     ^  the  12th8  is  17  twelfths,  or  1 

5     g  unit  and  5  twelfths  ;  we  write 

^  ^2  io  fractions'  column,  and 

carry  1  to  units*  column. 
If 

NoTB.  —  When  the  fractions  are  ready  for  addition,  the  work  appears  as  at 
the  left;  when  finished,  as  at  the  right.  In  practice,  it  should  not  be  begun  in 
one  place  and  finished  in  another. 

80.  Find  the  sum  of  48 J|,  32f^,  76f4. 

^rs     '^* 

?^^      160  ^^  =  ^tH  ;  H  =  m  ;  H  =  m-     The  sum  of  the 

tf^^j       iw  ^^^^      286ths  is  619  286ths,  or  2  units  and  49  285ths ;  we 

76—      tiS  write  49  285ths  in  fractions'  column,  and  carry  2  to 

^^ units'  column.     See  p.  20. 


OOMPUTID 

9 

s 

8 

S 

u 

9 

4 

n 
It 

..^U9       619 


Note.  —  It  is  convenient  to  express  the  L.  C.  D.  by  its  factors.  Thus :  the  L.  C.  D. 
is  1>5  X  3 ;  <)5  is  contained  in  this  3  times ;  57,  or  19  X  3,  is  contained  5  times ;  19  is 
contained  5  x  3,  or  15  times.    See  p.  73. 

Find  the  value  of: 

^    l+i;  A  +  i  +  l-  84.  98i  +  25f  +  5^. 

82.  i-f f;  i^ir  +  f  +  l.  85.  65f +  91f +  8f. 

83.  -1  +  ^;  H  +  i  +  f  86.  27J  +  18|  +  7f 


COMMON   FRACTIONS  89 

SUBTRACTION 


Before  fractions  can  be  subtrax^ted,  they  must 
be  reduced  to  equivalent  fractions  having  a  com- 
mon denominator. 


5^_9 
4    IB 
g  _  8 
The  least  common  denominator  should  always  s~J2 

be  found. 


87.  From  }  subtract  }. 


OOMPLBTCD 


9 

8 

Ig 

1 

IS 

»     g  leaves  ^  ;  we  write  ^  in  frac- 

^  tions'  column. 

7i 

Note.  —  When  the  fractions  are  ready  for  subtraction,  the  work  appears  as  at 
the  left;  when  finished,  as  at  the  right.  In  practice,  it  should  not  be  begun  in 
one  place  and  finished  in  another. 

8a  From  300^  subtract  200j^. 

^nn^  '""a  A  =  t!t  ;  1^1  =  4»(jV  ;  21  204ths  from  8  204th8  we 

*^  Hi  cannot  take  ;  we  add  204  204ths  to  8  204ths  making 

pnnL  ♦,  212  204ths,   and  1  to  0  units  making  1  unit;    21 

'-'H  ^^  **  ^      204th8  from  212  204ths  leaves  191  204tlis  ;  we  writ6 

qqIQI  191  1^^  204tfts  in  fractions'  column  ;  201  units  from  300 

iou  W4  units  leaves  99  units.     See  p.  SO. 

NoTK.  — It  is  convenient  to  express  the  L.C.D.  by  its  factors.  Thus:  the 
L.  C.  D.  is  .'il  X  4 ;  61  is  contained  fn  this  4  times ;  68,  or  17  X  4,  is  contained  3 
times.    iS«e  p.  73. 


Find  the  value  of: 

»   i-    I 

91.   J-    } 


i-i;H-f  92-  ^-n-'  68,V-27ii. 

i-i<  H-i         93-  H-n-'  75A-29H- 

|-»i  ji-f  94.  4}-2J;   67^-30?!. 


90 


MULTIPLICATION 


MULTIPLICATION 

Common  method 

Multiply  the  numerators  for  a  new 
numerator  and  the  denominators  for  a 
new  denominator,  canceling  wJien  pos- 
sible. 

Mixed  Jiumbers  should  be  reduced  to 
improper  froAUions, 


0     c 


}  of  AB  =  AC  ; 
§of  }of  AB  =  2  AC  =  AD 

J  of  AB  =  AD. 
.  f  of  }of  AB  =  i()f  AB, 

or  §  of  I  =  i. 


95.  Multiply  J  by  4. 

96.  Multiply  f}  by  5J  by  2\. 

Find  the  value  of: 
97.  Jx6;  8xt;  9xi. 
9a  }Xt\;  7x|;  9Xt3j. 
99.  Jxl6;  18  X  I;  72  x  J. 
100.  84  XtV;  |x96;  f  x  48. 


Divide  both  4  and  8  by  4  ; 
i.e.,  cancel  4  from  both  4  and 
8.    See  p.  72. 


6J  =  Y ;  ^=\^  cancel  7 
from  21  and  49  ;  7  from  7  and 
7  ;  4  from  36  and  4  ;  3  from  3 
and  3. 


102.  fof^fe;  }of  J;  y^oflG. 

103.  iof25;  f  of  24;  |of  16. 

104.  I  of  18;  ^x91;  |  x  65. 

105.  «of21;y9yOf  44;  ^7^  of  65. 


101.  96xy\;  |Jx84;  ^X64.    loa  |of20;  ^of  91;  t\  of  55. 


Find  the  value  of: 
107.  H  X  if  X  t^  X  A- 

lOa  5^  X  6}  X  6}  X  8|. 

109.  if  X  H  X  H  X  H- 
no.  f  X  f  J  X  1  Jj  X  II  X  «. 

Ill-  Ht  X  tt  X  H  X  «  X  H- 


112.  tt  X  ^V  X  tf  X  12f 

113.  15|  X  4f  X  8rV  X  -h' 

114.  55i  X  If  X  Hi  X  if 

115.  f  of  If  of  A  of  H  of  if. 

116.  l^x5Jx  Axff  xl^. 


COMMON  FRACTIONS 


91 


9     S_98th» 
8  ■  i~6«'A« 


Dividing  the  numerators 
and  the  denominators  gives 
tlie  same  result. 

9-4-3^3 

8  +  4     2 


DIVISION 
European  method 

Divide  the  numerators  for  a  new  nw- 
merutor  and  the  denominators  for  a  new 
denominatorf  changing  to  equivalent  frac- 
tions with  their  least  common  denomi- 
nator, if  necessary. 

Mixed  numbers  should  be  reduced  to 
improper  fractions. 

117.  Divide  }  by  |^  mentally. 

When  J  is  reduced  to  12th8,  tlie  numerator  be- 

^  _^  5  __  9_^  comes  9 ;  when  ^  is  reduced  to  12ths,  the  numer- 

j^  '  6      10  a^or  becomes  10 ;  9  h-  10  =  ^q.     The  quotient  of 

the  denominators,  12  -f-  12,  is  1. 

NoTB.  —  When  two  fractions  have  the  same  denominator,  since  the  quotient 

of  the  denominators  is  always  1,  it  is  necessary  to  think  of  the  numerators  only. 

lia   Divide  9J  by  lOf  mentally. 


9'-^  tO-  = 

75 

When  reduced  to  Sths,  the  numerator 

"«  ■  '"4 

86' 

becomes  73 ;  of  10|,  86 ;  the  answer  is  J  J. 

Find  the  value  mentatty: 

119-   i  +  f 

130. 

16 -Hf 

141. 

5*^3}. 

12a   f-t-J. 

131. 

18 -f 

142. 

2i^li. 

121.    i  +  }. 

132. 

27 -hf. 

143. 

^^^• 

122.   f-!-f. 

133. 

63-*-}. 

144. 

n-^H- 

123.    }-Hj. 

134. 

20-s-f 

145. 

7} +  8}. 

la*-  i  +  l- 

135. 

25 -J- f 

146. 

6|  +  8}. 

125.   t  +  f 

136. 

49 -hf 

147. 

9f  -h  6f 

126.   J  +  f 

137. 

84-*-}. 

14a 

^^^' 

127.   1  +  }. 

13a 

48 -hf 

149. 

8|^6i. 

12a    i  +  f 

139. 

54  H-  }. 

15a 

91-*- 3}. 

xaft  |  +  |. 

14a 

77  +  }. 

151. 

7^^2i, 

92 


DIVISION 
Common  method 


Invert  the  divisor  and  proceed       Inverting  the  divisor  and  proceed- 


as  in  multiplication. 


152.  Divide  }  by  4. 
8     4     32 


153.  Divide  4  by  J. 


ing  as  in  multiplication  gives  the  same 
result. 

}  X  f  =  |. 

15*.  Divide  J  by  }. 

8     8     64 

155.  Divide  16J  by  3}. 
10 

S 


Divide,  inverting  the  divisor: 

156.  if  by  18.  159.   36byT>p 

157.  Hbyie.  leo.   25  by  f 
15a  iV^y21.              161.  48  by  i  J. 

Divide,  inverting  the  divisor : 

165.   14tf  by41.  17a   113  by  13^. 


166.  17|4byl3. 

167.  18}fby33. 
16a  16||by79. 
169.   19ifby56. 


171.  594  by  18^. 

172.  190  by  26Jj. 

173.  850  by  38^. 

174.  221  by  583V 


1€2.  ttbyf 

163.  A  by  H- 

16*.  HbyA- 

175.  42fby36yV 

176.  57f  by  12^ 

177.  If2|  by  ^, 

17a  2^byfH 

179.  2^  by  ^. 


COMMON   FRACTIONS  93 

SPECIAL  METHODS 

laa  Multiply  475  by  361. 

475 

gQ*  gjjo                We  first  multiply  476  by  J  ;  2  times  476  = 

I  950 ;  950  ^  3  =  3l«j.     It  is  well  to  write  950 

316-  to  the  right  as  we  multiply,  and  to  divide  by  3 

aosn  without  writing  the  3. 

"^^^^  We  then  multiply  476  by  36  in  the  usual 
way. 


1425 

17416'- 


laL  Multiply  638f  by  27i. 


638l      ^*'^ 
J7|        SI 


4^51        U  We  first  multiply  J  by  | ;  then,  638  by  §  ; 


4466 
1276 


^J\  then,  }  by  27  ;  then,  638  by  27. 


17672j-^     S 

182.  Divide  65J  by  4. 

/  \  j^?  6  tens  -f-  4  =  1  ten  and  1  ten  remaining ;  we 

I £.  write  1  in  tens'  column. 

IS—  1  ten  and  5  units  =16  units ;   16  units  -f- 

''  4  =  3  units  and  3  units  remaining,  we  write  3 

in  units'  column. 
3  unita  and  2  thirds  =  Y ;  V  •<-  ^  =  H-     ^^^  P-  ^i- 

183.  Divide  275  by  3J. 

_f  \  ^j-  Multiplying  both  dividend  and  divisor  by  the 

o-)4io  'game  number  does  not  affect  the  quotient. 

11)825  ^^  multiply  both  dividend  and  divisor  by 

^j'  something  that  will  make  the  divisor  an  in- 

teger.   Here,  we  multiply  both  terms  by  3. 


94 


i 

m 

THE  FOUR 

OPEllATIONS 

Add: 

THE  FOUR 

OPERATIONS 

184. 

185. 

186. 

187. 

23A 

14A 
12A 

m 

7A 

13A 
ISA 

12H 

Subtract: 

18a 

189. 

190. 

191. 

263Hi 

7345», 
673H 

325A 
266H 

192. 

193. 

194. 

195. 

145}J 
122H 

164^1 
142Hi 

585H 
425^ 

424tt 
222H 

Multiply: 

196. 

197. 

19a 

199. 

324J 
48 

425 

75f 

625f 
28| 

738| 
2311 

200. 

201. 

202. 

2oa 

687f 
27i 

584J 
48| 

938f 
69^ 

865| 
36f 

Divide : 

204. 

205. 

206. 

207. 

4)723^ 

3)9345i 

4)6785| 

7)92861 

2oa 

209. 

210. 

211. 

4i}935 

62^)3863^ 

4281)3285 

2661)3586 

COMMON  Fractions 


95 


COMPLEX  FRACTIONS 

To  simplify  complex  fractions,  it  is  often  best  to  multiply 
both  terms  by  the  least  common  denominator  of  the  several 
fractions. 


212. 

Simplify  1^. 

Tkis  =  l±^,  =  f 

2ia 

Simplify  ^. 

214. 

Simplify  i^. 

I 

5        g 

215. 

Simplify  j*~*. 

ThaA  =  S. 

87 


216.   Simplify  1^1^-1 

2%w  =  f  xjx|xjx 
«x?  =  l. 


The  L.  C.  D.  of  the  several  frac- 
tions is  12;  12xi=6;  12x^  =  4; 
12  X  f  =  9 ;  12  X  i  =  2. 


The  numerator  of  the  complex  frac- 
tion is  f  ;  the  L.  C.  D.  of  f ,  J,  and  | 
is  15. 


Before  applying  this  principle, 
compound  terms  must  be  simplified. 
|x|  =  i. 


When  the  L. CD.  is  lai^e,  it  is 
often  better  to  perform  the  indicated 
operations. 


If  a  fraction  is  used  an  odd  num- 
ber of  times  as  a  divisor,  we  invert ; 
if  an  even  number  of  times,  we  do 
not  invert. 


NoTB.  —  In  Ex.  214,  the  L.C.D.  of  the  simple  fractions  is  12.  If  we  had  mnl- 
tiplied  both  i  and  ]  by  12,  the  numerator  would  have  been  multiplied  by  12  x  12, 
or  144.    See  p.  72. 


96  COMPLEX   FRACTIONS 

It  should  not  be  forgotten  that  complex  fractions  may  always 
be  solved  by  performing  the  indicated  operations. 

217.  Solve  examples  212,  213,  216  by  performing  the  indicated 
operations. 

2ia  Solve  again  example  216,  as  on  page  95,  and  explain  why 
J  is  inverted ;  J,  not  inverted ;  J,  inverted ;  f ,  not  inverted. 

I  is  used  once  as  a  divisor  (it  is  in  the  denominator  of  tlie  complex  frac- 
tion), invert;  |,  twice  (preceded  by  ♦-,-,'  and  in  the  denominator  of  the 
complex  fraction),  do  not  invert;  J,  once  (the  second  complex  fraction  is 
preceded  by  •-*-'),  invert;  ^,  tteice  (tiie  whole  complex  fraction  is  preceded 
by  '-{-,*  and  ^  is  in  the  denominator),  do  nut  invert. 


Using  expedients  of  page  95,  simplify : 


219. 

1. 

V 

i 
* 

22a 

V 

i. 

223. 

± 

x3|. 

224. 

^ 

xlf. 

h- 

■A 

*- 

-i 

99fl 

n 

-1 

3i 

-2| 

229. 

54 

x8i 

^ 

^tt 

222  li;  5i. 


225.  gxf 


230.   -r# — ^X- 


231. 


232. 


6|of3;^-2f 


COMMON   FRACTIONS  97 

MISCELLANEOUS 

233.  Analyze  aud  read  |  by  the  tirst  conception ;  by  the  second. 

234.  Give  the  meaning  of  |  by  the  first  conception;   by  the 
•  coikI. 

235.  Why  is  J  not  a  fraction  by  the  second  conception  ?  Why 
is  it  a  fraction  by  the  first  conception  ? 

236.  Reduce  -j^y  to  lowest  terms ;  state  the  principle,  and  the 
object  of  the  reduction. 

237.  Reduce  J|4  to  lowest  terms,  and  explain  how  we  proceed 
when  no  common  factors  can  be  found  by  inspection. 

23a  Change  f  to  24ths ;  state  the  principle.  Why  do  we  re- 
duce fractions  to  higher  terms  ? 

239.  Change  |J  to  a  whole  or  a  mixed  number ;  explain.  What 
is  the  object  of  this  reduction  ? 

240.  Reduce  6|  to  an  improper  fraction ;  explain  in  two  ways. 

241.  Why  are  whole  and  mixed  numbers  reduced  to  fractions  ? 

242.  Add  and  explain :  22 j,  16J ;  subtract  and  explain ;  multi- 
ply and  explain  ;  divide  and  explain ;  find  the  product  of  the 
sum  and  the  difference. 

243.  Multiply  632|  by  24} ;  (a)  by  the  common  method ;  (6)  by 
the  special  method.     Which  do  you  prefer  ? 

244.  Divide  }  by  f ;  (a)  by  the  European  method ;  (b)  by  the 
common  method.     Which  do  you  prefer  for  mental  work  ? 

245.  Divide  157J  by  6 ;  (a)  by  the  common  method ;  (6)  by  the 
special  method.     Which  do  you  prefer  ? 

246.  Simplify  t-Hl ;  (a)  by  performing  the  indicated  opera- 
tions ;  (6)  by  the  expedient  on  page  95.     Which  do  you  prefer  ? 

247.  Simplify  | — v|  -h  ^^  ;  (a)  by  performing  the  indicated 
oi)erations ;  (b)  by  the  expedient.     Which  do  you  prefer  ? 

AMBS.    ARITH. — 7 


98  ANALYSIS 

ANALYSIS 

The  pupil  will  derive  great  benefit  from  writing  out  the  rela- 
tions. 

State  the  relations  and  analyze: 

24a   If  1  yard  of  cloth  costs  12^,  how  much  will  5^  yards 

cost? 

Belation :  cost  5J  yards  =  6^  times  cost  1  yard. 

Analysis:  since  1  yard  costs  12^,  5^  yanis  will  cost  6}  times  12^,  or 
mf. 

Note.  —  Some  prefer  two  relation* :  cost  5  yards  =  6  times  cost  1  yard ;  cost 
I  yard  =  icost  1  yard.  Analysis :  the  cost  of  5  yards  is  5  times  12  cents,  or  60 
cents ;  the  cost  of  I  yard  is  I  of  12  cents,  or  6  cents ;  the  cost  of  5k  yards  is  the 
sum,  or  66  cents. 

249.  If  I  of  a  yard  of  cloth  costs  12^,  how  much  will  1  yard 

cost? 

Belation :  cost  1  yard  =  j  cost  }  yard. 

Analysis:  since  |  of  a  yard  costs  12/»,  1  yard  will  cost  |  of  12^,  or  16^. 

Note.  —  Some  prefer  two  relations:  cost  of  i  of  a  yard  =  i  cost  of  J  yard; 
cost  of  I  =  4  times  cost  of  J.  Analysis:  since  i  of  a  yard  costs  12  cents,  i  of  a 
yard  will  cost  k  of  12  cents,  or  4  cents ;  |,  or  1  yard,  will  cost  4  times  4  cents,  or  16 
cents, 

250.  15  is  f  of  what  number  ? 

Belation :  the  required  number  is  J  of  the  given  number. 
Analysis :  since  J  of  a  number  is  16,  the  number  is  f  of  15,  or  20. 

Note.  — Some  prefer  two  relations:  k  of  the  number  =  J  of  }  the  number;  J, 
or  the  number,  =  4  X  i  the  number.  Analysis :  since  J  of  a  number  is  15,  i  of  the 
number  is  i  of  15,  or  5 ;  },  or  the  number,  is  4  times  5,  or  20. 

251.  If  1  bushel  of  potatoes  costs  36  ^,  how  much  will  2f  bush- 
els cost  ? 

252.  If  2^  bushels  of  apples  cost  90  ^,  how  much  will  1^  bushels 
cost? 

253.  20  is  f  of  what  number?  30  is  f  of  what  number?  45 
is  f  of  what  number  ? 

254.  A  watch  was  sold  for  $  12.  The  selling  price  was  f  of 
the  cost.     What  was  the  cost  ? 


COMMON    FRACTIONS  ;;1J^  ; 

255.  A  man  lost  ^\  of  liis  money  and  afterwards  found  |  of 
what  he  had  lost.    What  part  of  the  original  amount  had  he  then  ? 

256.  A  man  bequeathed  ^  of  his  estate  to  his  wife,  J  of  the 
remainder  to  his  daughter,  and  the  part  remaining  to  his  son. 
What  part  of  the  estate  did  the  son  receive  ? 

257.  In  a  certain  school,  f  of  the  scholars  belong  to  the  fourth 
class,  J  to  the  third,  |  to  the  second,  and  the  remainder  to  the  first 
class.     What  part  of  the  school  belongs  to  the  first  class  ? 

25a  Of  a  farm  of  160  acres,  f  is  used  for  grazing,  f  for  corn, 
^  for  wheat,  and  the  rest  for  oats.  How  many  acres  of  oats 
are  there  ? 

259.  If  IJ  yards  of  cloth  are  required  for  each  coat,  how  many 
yards  will  be  required  for  17  coats  ? 

260.  How  many  coats  may  be  made  from  33|  yards  of  cloth, 
if  1 J  yards  are  required  for  each  coat  ? 

261.  Of  a  farm  of  160  acres,  48  acres  are  in  wheat,  36  acres  in 
oats ;  the  remainder  is  pasture.  What  part  of  the  whole  is  used 
for  pasture  ? 

262.  If  my  whole  farm  is  planted  with  corn,  wheat,  and  oats, 
what  part  of  the  whole  is  in  corn,  if  I  have  twice  as  many  acres 
of  corn  as  of  wheat,  and  three  times  as  many  acres  of  oats  as  of 
corn  ? 

263.  A  grocer  sells  18  bunches  of  radishes  at  4|^  a  bunch,  12 
quarts  of  peas  at  2}^  a  quart,  and  takes  his  pay  in  eggs  at  13^^  a 
dozen.     How  many  dozen  eggs  does  he  receive  ? 

264.  Henry  is  18  years  old,  or  }  as  old  as  Albert.  How  old  is 
Albert? 

265.  A  farmer  sold  a  quantity  of  rye  for  $  24,  which  was  only 
J  of  its  value.     How  much  did  he  lose  ? 

266.  If  j  of  a  ship  is  worth  $  12,000,  how  much  is  i  of  it  worth  ? 

267.  Jane  spent  30  cents,  or  jf  of  her  money,  for  a  book ;  with 
the  remainder  she  bought  apples  at  2^  apiece.  How  many 
apples  did  she  buy  ? 


IOC  ANALYSIS 

26a  If  1  yard  of  cloth  costs  $  5J,  how  much  will  12J  yards 
cost? 

269.  If  4  yards  of  cloth  cost  99^,  how  much  will  }  of  a  yard 
cost  ? 

27a  A  lady  gave  \  of  all  her  money  for  a  dress,  and  \  of  it  for 
a,  shawl.     What  part  remained  ? 

271.  A  farmer,  having  lost  24  sheep,  had  only  J  of  his  floik 
remaining.     How  many  sheep  had  he  at  first  ? 

27X  If  6  apples  cost  7^,  how  many  apples  can  be  bought 
for  14^? 

273.  B  owned  |  of  a  ship,  and  sold  }  of  his  share.  What  part 
of  the  whole  ship  did  he  still  own  ? 

274.  )  of  6  is  }  of  what  number  ?    J  of  10  is  j  of  what  number  ? 

275.  If  12  apples  cost  2^  ^,  how  many  apples  can  be  bought  for 
26^? 

276w  A  coat  cost  $  20,  and  f  of  the  cost  of  the  coat  is  ^  of  the 
price  of  the  suit.     What  is  the  price  of  the  suit  ? 

277.  A  farmer  has  J  of  his  cattle  in  one  field,  J  in  a  second, 
and  the  remainder,  or  15  head  of  cattle,  in  a  third.  How  many 
cattle  are  in  the  herd  ? 

27a  If  I  of  a  yard  of  silk  costs  }  of  a  dollar,  how  much  will 
I  of  a  yard  cost  ? 

279.  A  is  f  as  old  as  B ;  B  is  f  as  old  as  C ;  C  is  60  years  old. 
How  old  is  A  ? 

280.  If  the  difference  between  }  and  j  of  my  age  is  5  years, 
how  old  am  I  ? 

2aL  If  7  men  can  do  a  piece  of  work  in  4^  days,  how  long  will 
it  take  6  men  to  do  the  same  work  ? 

282.  If  I  give  18|  bushels  of  potatoes  at  60  cents  a  bushel  for 
cloth  at  22J  cents  a  yard,  how  many  yards  of  cloth  do  I  receive  ? 

283.  If  a  man  can  do  a  piece  of  work  in  9  days  by  working  7J 
hours  i^er  day,  iu  how  many  days  of  8J^  hours  each,  can  he  do  the 
same  work? 


COMMON  FRACTIONS  101 

PROOFS 

I.  Multiplying  the  numerator  multiplies  a  fraction. 

Multiplying  the  numerator  multiplies  the  number  of  equal  parts  that  are 
Uiken,  without  affecting  the  size  of  the  parts,  and  thus  multiplies  the  frac- 
tion. 

I I.  Multiplying  the  denominator  divides  a  fraction. 

Multiplying  the  denominator  multiplies  the  number  of  equal  parts  into 
which  the  unit  is  divided,  thereby  dividing  the  size  of  each  part,  without 
affecting  the  number  of  parts  taken,  and  thus  divides  the  fraction. 

III.  Dividing  the  numerator  divides  a  fro/ction. 

Dividing  the  numerator  divides  the  number  of  equal  parts  that  are  taken, 
without  affecting  the  size  of  the  parts,  and  thus  divides  the  fraction. 

IV.  Dividing  the  denominator  multiplies  a  fraction. 

Dividing  the  denominator  divides  the  number  of  equal  parts  into  which 
the  unit  is  divided,  thereby  multiplying  the  size  of  each  part,  without  affect- 
ing the  number  of  parts  taken,  and  thus  multiplies  the  fraction. 

V.  Multiplying  both  numerator  and  denominator  by  the  same 
number  does  not  change  the  value  of  a  fraction. 

Since  multiplying  the  numerator  multiplies  the  fraction,  and  multiplying 
the  denominator  divides  the  fraction,  multiplying  both  terms  by  the  same 
number,  first  multiplies  and  then  divides  the  fraction  by  the  same  number, 
and  does  not,  therefore,  change  the  value  of  the  fraction. 

VI.  Dividing  both  numerator  and  denominator  by  the  same  iium- 
ber  does  not  change  the  value  of  a  fraction. 

Since  dividing  the  numerator  divides  the  fraction,  and  dividing  the  denomi- 
nator multiplies  the  fraction,  dividing  both  terms  by  the  same  number,  first 
divides  and  then  multiplies  the  fraction  by  the  same  number,  and  does  not, 
therefore,  change  the  value  of  the  fraction. 

Note.  —The  pupil  should  distinguish  carefully  the  difference  between  proof  $ 
and  illustrations.    See  pp.  86,  86,  90, 91. 


102  PROOFS 

VII.    To  multiply  fractionSf  multiply  the  numerators  for  a  new 
numerator  and  the  denominators  for  a  new  denominator. 

To  prove  that  -  x  ^  =  ^JSA, 

*^  S      5     SxS 

?  X  4  -  ^_ ^_j.  {MuiUplying  th»  numtra 

3  3     *  tor  tnuttiplie»tht  fraction.) 

Since  4  is  6  x  ^  inuliiplying  by  4  is  multiplying  by  a  number  5  times  too 
large ;  tlie  product  in  6  times  too  large,  and  must  be  divided  by  5. 

2x4  ^  5  _  2x4  {MHltiptying  tks  denomi- 

3  3x6  nator  dttidet  tke/raettom.) 

. 2       4^2x4 
"3      6     3x6* 

VI IT.    To  divide  frcuiionSf  divide  the  numerators  for  a  new 
numerator  and  the  denominators  for  a  new  denominator. 

Toprovethat  *+?  =  ^j!l?. 

^  9      3     9+3 

8       ft  _  8  •t-2  (Dinidtng  Vu  numerator 

g"*"  ~9  divid0$  tk4  fraction.) 

Since  2  is  3  times  f,  dividing  by  2  is  dividing  by  a  number  3  times  too 
large ;  the  quotient  is  3  times  too  small,  and  must  be  multiplied  by  3. 

8  -i-2  J.  3  _  8  -s-2  (Diridinff  the  denominn- 

9  9  T-  3  tor  muUiplies  ths fraction.) 

.    8  ^  2_8-^-2 

IX.    To  divide  fractionSy  invert  the  divisor  and  proceed  as  in 
multiplication, 

Toprovethat  f  +  ?  =  ^x^- 

9     3     9     2 


8    ,    o  _      8  (Multiplying  the  detiomi- 

Q  "^  ^  ~Q       9*  naior  divides  Ote  fraction.) 

Since  2  is  3  times  |,  dividing  by  2  is  dividing  by  a  number  3  times  too 
large ;  the  quotient  is  3  times  too  small,  and  must  be  multiplied  by  3. 

8      w  *>  _  8x3  (Multiplying  the  numera- 

9x2  "9x2*  tor  multiplies  Vie  fraction.) 

,  8^2_8^3 
•9     3-92 


DECIMALS 


TERMS  AND  RELATIONS 


A  common  fraction  may  have  for 

its  denominator  10,  100,  1000, 

a  decimal  fraction. 

A  decimal  fraction  may  be  re- 
solved into  a  series  of  decimal  frac- 
tions whose  denominators  are  10, 
100,  1000,  ....  and  whose  numera- 
tors are  less  than  ten. 

Since  ....  10  thousandths  make 
1  hundredth,  10  hundredths  make  1 


tenth,  . 


this  series  of  fractions 


may  be  expressed  by  the  devices  of 
the  Arabic  notation.     See  p.  10. 

By  the  Arabic  notation,  the  names 
of  the  orders  are  expressed  by  rela- 
tive position.  Thus,  378  means  3 
hundreds  7  tens  8  units.  If  378  is 
to  mean  3  tentfis  7  hundredtJis  8 
thousandthsy  an  additional  device  is 
necessary.  This  device  consists  in 
placing  before  tenths  a  period,  called 
the  decimal  point. 

A  decimal  fraction  whose  denomi- 
nator is  expressed  by  a  decimal  point, 
is  a  decimal. 

103 


Illlstrations 
^,/\y,  a  decimal  fraction. 

Since  378  =  300-1-70-1-8, 


jhh  —  T^^yi   iViJ  —  !*<»» 


Write  T^5,  using  the  de- 
vices of  the  Arabic  notation. 

S7S  _S_,_7_        S 
1000     10     100     1000 

S  tenths  7  hundredths  8  thou- 
sandths, decimal  378, 

.378 


.378,  a  decimal. 


104  TERMS   AND   RELATIONS 

Write  the  steps  to  show : 

1.  ThatVV(F  =  .37  3.  That  T^HHhr  = -0379 

2.  That  tI J^=.a37  4.  That  ,Tft^  = -0037 

Ex.  2.  37  =  000  +  30  +  7 ;  jUn  =  A%%  +  dh  +  lo'ou  =  A  +  lU  +  TO^ffj 
=  0  tenMa  3  hundredths  7  thousandths  =  decimal  037  =  .037. 

Write  the  steps  to  show  : 

5.  That  .93  =  T^.  7.  That  .0973  =  ,^Jt^. 

6.  That  .093  =  T^^f^.  a  That  .0097  =  y^ftVr. 

Ex.  6.  .003  =  0  tetUhs  0  hundredths  3  thousandths  =  i^«  +  ^U  +  r/sv 
=  jlh  +  tAb  =  r!l5- 

9.  State  the  numerator  of  each  decimal  in  examples  1  to  8, 
and  observe  that  it  is  the  decimal  regarded  as  an  integer. 

Ans.  Ex.  4.  In  .00:37,  or  xo'oVo*  ^^^  numerator  is  37,  or  it  is  the  decimal 
.0037  regarded  as  an  integer. 

10.  State  the  denominator  of  each  decimal  in  examples  1  to  8, 
and  observe  that  it  may  be  ascertained  by  numerating  from  the 
decimal  point  to  the  right-hand  figure. 

Ans.  Ex.  4.  In  .0037,  or  roVoof  ^^®  denominator  is  10,000.  Numerating 
.00^37  ;  tenths,  hundredths,  thousandths,  ten-thousandths. 

11.  Show  that  the  decimal  orders  are:  tenths,  hundredths, 
thousandths,  ten-thousandths,  hundred-thousandths,  millionths,... 

Ans.   Y^  =  ^  ;  yjj  =  jl^jf ;  y^j,^  =  ji^^gn  ;  ti^H^  =  Tiri^uff  J  •  •  • 

12.  Name  the  decimal  orders  to  quadrillionths ;  give  the  table 
of  decimal  orders  to  quadrillionths.     See  p.  11. 

13.  Show  that  the  decimal  periods  are :  thousandths,  millionths, 
billionths,  trillionths,  quadrillionths,  ....     Seep.  11. 

Ans,    y^rtnT  ~  TooWW »    lOOoOOO  —  1000000000  i   •    •    •    • 


DECIMALS  105 

NUMERATION 

Read  a  decimal  as  a  common  froL'tion;  first  its  numeratOTy  then 
its  denominator  in  the  ordinal  form. 

Read  the  numerator^  the  denominator,  the  fraction  : 

14.  .5;  .06;  .007  17.   .0002006;  .00000213 

15.  .16;  .027;  .0039  la  .003080654 

16.  .368;  .0407;  .00362       19.  .0123005678937286 

Ex.  18.  'Hie  numerator  is  3,080,654  ;  the  denomiuator,  1  billion  ;  the  frac- 
tion, 3,080,654  billion ths. 

NoTK.  —  In  Ex.  19,  we  may  numerate  by  periods;  thousandths,  millionths, 
billlonths,  trillionths,  quadrillionths,  ten-quadrillionths. 

A  mixed  decimal  is  made  up  of  an  integer  and  a  decimal. 

•    Analyze  and  read  : 

20.  175.68;  400.006;  .406  23.  23.0708;  35682.3 

21.  3000.003;  20.586;  1.7  24.  7583.64;  700.0075 

22.  1.0003;  62.832546  25.  328.00062;  10.0001 

Ex.  20.  175  is  the  integer  ;  .68,  the  decimal ;  read,  175  and  68  hundredths. 

NoTK.  —  In  reading  mixed  decimals,  the  word  "and  "should  be  used  at  the 
decimal  point,  but  nowhere  else.  See  p.  9,  Note.  Without  this  understanding. 
/our  hundred  and  six  thoitsandths  might  be  written  either  400.006,  or  .406. 

A  complex  decimal  is  made  up  of  a  decimal  and  a  common 
fraction.  A  common  fraction  can  not  occupy  a  decimal  order,  but 
is  of  the  same  denomination  as  the  order  which  it  follows. 

Read: 

26.  .061;  -^H  2a   .OOi;  .OJ;  .^ 

27.  .0008i;  .OOOJ  29.   20.086J;  5.0123| 

Ex.  26.  Ans.  6^  hundredths;  1}  Ex.  28.  Am.   .|  is  an  improper 

thousandths.  expression. 


106 


WUIATION 
ROTATION 


Write  the  numercUor  as  in  common  fractions,  and  pface  the  deci- 
mal point  so  as  to  make  the  name  of  tfie  last  order  the  name  of  tlie 
denominator. 

aa  Write  638  millionths. 

,000638  We  write  the  numerator  as  in  common  frac- 

tious, 638 ;  looking  at  8  we  thinlc  millionths 
because  this  is  tlie  name  of  the  denominator  ;  at  3,  hundred-thousandths ;  at 
0,  ten-thousandths;  supplying  0  (it  now  appears  0038),  we  tiiinlc  tlitm- 
sandllis;  supplying;  0(00038),  we  think  hundredths;  supplying  0(000038), 
we  think  tenths  ;  writing  the  decimal  point,  we  have  .000038. 


Write  decimally  : 

31.  Five  hundred  millionths. 

32.  Seventeen,  and  7  tenths. 
3a  (>38  hundred-millionths. 

34.  Twenty-three  trillionths. 

35.  85  thousand  2  billionths. 

36.  7007  ten-thousandths. 

37.  3G0  hundred-thousandths. 


3a  Six  hundred  four  thousandths. 
39.  9431,  and  906532  ten-milliontli 
4a  Two  thirds  ten-thousandths. 
4L  2  billion,  and  two  billionths. 
42.  83  thousand,  and  5  thousandtlis. 
4a  99  thousand  999  ten-millionths. 
44.  1  million  6  thousand  4  billionths. 


Read: 

45. 

75.0G432 

4e 

.000000} 

47. 

35.02767 

4a 

999.9999 

49. 

9.0999999 

50. 

200.0016 

51.   1400.014567 


52. 

62.00634010 

53. 

.09022635 

54. 

304.00672 

55. 

.98637208 

56. 

.83000008 

57. 

9200.0929 

5a 

9000.0009372 

DECIMALS  107 

ADDITION 
Write  units  of  the  same  order  in  the  same  column. 

59.  Add  36.03,  7.864 

36.03 
7.864.  ^^^  explanation,  see  p.  SO.    The  sum  of  the 


43.894 


tboiuiandths  is  4  thousandths :  etc. 


SUBTRACTION 
Write  units  of  the  same  order  in  the  same  column. 

60.  From  38.03  subtract  25.128 

38.03 

S5.128  ^^^  explanation,  see  p.  SO.   8  thousandths  from 

'. 0  thousandths  we  cannot  take  :  etc. 

1^.90£ 

MULTIPLICATION 

Multiply  as  in  integers,  and  point  off  as  many  decimal  places  in 
the  product  as  there  are  decimal  places  in  both  multiplicand  and 
multiplier. 

61.  Multiply  .473  by  .23,  and  explain. 

.473  To  multiply  fractions,  we  multiply  the  numera- 

09  tore  for  a  new  numerator,  and  the  denominators 

! —  for  a  new  denominator.     See  p.  90. 

1419  The  numeratore  are  47.3  and  2.3  ;  we  multiply 

giQ  them  as  in  integere.    The  denominatore  are  KXK) 

and  100 ;  their  product  is  100,000  ;  we  point  off 


.10879  3  +  2,  or  5  decimal  places. 

62.  Multiply  378.6954  by  1000. 

^  Moving  the  decimal  point  one  place  to  the  ripht 

1000  multiplies  by  10  ;  two  places,  by  100 ;  etc.    See 


378695.4  P' 40,  Ex.  SI. 


108  DIVISION 

DIVISION 

Divide  as  in  integers,  and  point  off  as  7naiiy  decimal  places  in  the 
quotient  as  those  in  tJie  dividend  exceed  those  in  the  divisor. 

63L  Divide  .0414  by  .23  and  explain. 

£S).04J4(.1S  To  divide  fractions,  we  divide  the  numera- 

•^  tors  for  a  new  numerator  and  the  denominators 

jgi  for  a  new  denominator.     See  p.  91. 

2g,  I'lte  numerators  are  414  and  23;  we  divide 

^  them  as  in  integers.    The  denominators  are 

10,000  and  100  ;  their  quotient  is  100 ;  we  point  off  4-2,  or  2  decimal  places. 

GC  Divide  426  by  .08. 

Before  dividing,  it  is  necessary  to  annex 

.08)  4x6.00  ciphers  to  tlie  dividend  until  the  number  of 

5S25  decimal  places  equals  or  exceeds  those  in  the 

divisor. 

Since  tliere  are  two  places  in  the  divisor  and  none  in  the  dividend,  we 

annex  two  ciphers  to  the  dividend. 

65.   Divide  8%  by  432  true  to  two  decimal  places. 

432)896.00(2.07^^  ^  .  ^  ,     .     , 

86A  ®  annex  two  ciphers  to  make  the  places  in 

— — - —  the  dividend  exceed   those  in  the  divisor  by 

3024  ^1  should  be  reduced  to  lowest  terms. 

176 


NoTF.  —  If  the  exact  qnotient  is  not  required,  it  is  customary  to  write  '  + ' 
instead  uf  the  common  fraction. 

66.  Divide  268  by  10,000. 

10000)  268  Moving  the  decimal  point  one  place  to  the 

Q^0g  left  divides  by  10  ;   two  places,  by  100 ;  etc. 

See  p.  55,  Ex.  152. 

67.  Divide  .06  by  6000. 

6.000x).000it06  If  the  divisor  ends  with  ciphers,  we  move 

QQQQl  the  decimal  point  in  both  dividend  and  divisor. 

See  p.  55,  Ex.  15S. 


DECIMALS 


109 


THE  FOUR  OPERATIONS 


Add. 


6a   $988,058,  $75,896,  $75,  $68,299,  $86.43,  $45,256. 

69.  $77,948,  $89.23,  $57,637,  $88,009,  $6,783,  $.086,  $55. 

70.  $  235.06,  $  578.085,  $  735.88,  $  967.08201,  $  3.6872,  $  20.98, 
$66,  $8.7963,  $48.23415,  $  9996.6,  $  8763. 


Subtract  : 

71.  35.1789  from  103.45 

72.  .463268  from  13.603 

73.  21.9875  from  86.013 

74.  178.369  from  10,000 

Multiply : 

79.  3.876  by  .3816 

80.  .4081  by  1001 

81.  .6375  by  .5082 

82.  700.87  by  .642 

Divide  : 

87.  .00072  by  .016 

8a  63.904  by  4.86 

89.  .97230  by  .313 

90.  27013  by  176 

91.  17.181  by  567 

92.  5.0000  by  789 

93.  1.4511  by  .214 


75.   .166  from  55.03 
7a  8.954  from  9.862 
77.  .88875  from  100.02 
7a  88.86  from  856.9 

83.  23.478  by  3300 

84.  2.0087  by  2500 

85.  6.0961  by  1600 
sa  365.08  by  68.092 

94.  .0005  by  5000 

95.  5000  by  .0005 

96.  .1265  by  7.91 

97.  .9614  by  .907 
9a  .9396  by  67.12 
99.  6800  by  .0034 

100.  .0006  by  3400 


XoTR.  —  The  pupil  should  turn  to  p.  54  and  solve  examples  141  to  145,  carry 
ing  out  each  quotient  two  ur  three  decimal  places. 


110  COMMON   FRACTIONS  TO   DECIMALS 

COMMON   FRACTIONS   TO   DECIMALS 

101.  Reduce  |  to  a  decimal. 

8)6.000  I  means  6  -»-  8 ;  performing  the  indicated  oper- 

,625  atioD,  we  obtain  .625. 

102.  Reduce  ^  to  a  decimal. 

7)S.OOOOOO  Oar  laai  remainder  is  2,  and  the  next  dividend 

001:71  li  ^  ^»  ^*"^  ^^  ^*^  *^  fi"'-     Hence,  the  quotiem 

.^iS0il4^  will  repeal  the  figures  2,  8,  6,  7,  1,  4,  and  will  not 

ooiue  out  exact.    It  is,  therefore,  impossible  to  reduce  f  to  a  decimal  exactly. 

I.  Result  exact 

7)2.000  If  the  exact  result  is  required,  it  is  customary 

^  ^B  to  carry  out  the  division  a  few  places,  and  to  write 

-^^7  the  remainder  as  in  I. 

II.   Result  approximate. 

7)2.000  If  an  approximation  is  desired,  In  place  of  the 

JSSS-k-  remainder,  we  write  *  +  *  as  in  IL 

III.  Result  as  a  repetend. 

7)2.000000  M  we  desire  to  show  the  figures  which  repeat, 

9>JiK7lL  ^'^  carry  out  the  division  until  the  quotient  begins 

JSifOfi4  ^  repeat,  and  place  dots  over  the  first  and  last 

figures  of  the  repetend,  as  in  III.     The  result  is  a  circulating  decimal;  it  is 

read:  decimal,  repetend,  2,  8,  5,  7,  1,  4,  repetend. 

loa  What  fractions  cannot  be  reduced  to  decimals  exactly  ? 

An3.  Those  fractions  which  have  prime  factors  other  than  2  or  5  in  their 
denominators.  The  factors  of  10  are  2  and  6.  Hence,  if  a  fraction  has  any 
prime  factor  other  than  2  or  6  in  its  denominator,  the  reduction  will  not  be 
exact. 

104.  Can  y|y  be  reduced  to  a  decimal  exactly  ?    Why  ? 

105.  If  ^^  is  reduced  to  a  decimal  exactly,  how  many  decimal 
places  will  there  be  in  the  result? 

Ans.B,    ,i,=ixjxixjxixixjxi  =  (.6)». 


DECIMALS  111 

DECIMALS   TO   COMMON   FRACTIONS 

106.  Reduce  .G25  to  a  common  fraction. 

.625  =  AVff  =  g-  ^®  write  the  decimal  as  a  common  fraction, 

and  reduce  to  lowest  terms. 

107.  Reduce  .83J  to  a  common  fraction. 

§^  =  —  =  5.  We  first  reduce 

100      300      6         to  a  simple  fraction. 


.831  =  §^  =  ?^  =  ?.  We  first  reduce  the  complex  fraction,  ^ 


Reduce  to  common  fractions  in  lowest  terms : 

loa  .032;  .0016  115.  .12^;  .33J 

109.  .075;  .0128  11&   .87^;  .37f 

ua  .025;  .1126  117.   .83J;  .62^ 

111.  .005;  .3125  lia   .125f;  .0375 

112.  .004;  .1728  119.   .0875;  .OOOJ 

113.  .225;  .4375  120.   .086};  .216J 

114.  .024;  .1626  121.  .1875;  .080J 

Reduce  to  decimals  exactly : 

122.    i,    J                                125.    i,    J  12a    .75i 

12a    i,    i                               126.    ^,    ^  129.    .871 

124.    h    i                              127.    iVV»    tW  130.    .OO^V 

131.  Can  jj  be  reduced  to  a  decimal  exactly  ?  Why  ? 

132.  Reduce  jj  to  a  decimal,  writing  the  divisor  under  the 
remainder  after  the  third  decimal  place  of  the  quotient. 

133.  Reduce  fi  to  a  decimal,  writing  *  -f  '  after  the  fourth  deci- 
mal j)lace  of  the  quotient.  • 

134.  Reduce  H  to  a  circulating  decimal,  writing  dots  over  the 
first  and  last  figures  of  the  repetend. 


112  COMPLEX   DFXIMALS 

COMPLEX  DECIBIALS 

If  a  complex  decimal  is  to  be  subjected  to  any  operation,  it  is 
best  first  to  reduce  to  a  simple  decimal. 

135.  Prepare  36.00},  7.1}  for  the  operations. 

S6.00i  =  36.0025;    TJ^    =    7.1S75 

When  a  complex  decimal  cannot  be  reduced  to  a  simple  deci- 
mal, it  is  generally  sufficient  to  carry  out  to  three  or  four  decimal 
places  and  to  substitute  *  -f  ^  for  the  common  fraction. 

136.  Prepare  16.2|,  48.32}  for  the  operations. 

16.2\^  16,2S3^",    48.S2\  =  48,328-^- 

If  absolute  accuracy  is  required,  the  common  fraction  cannot 
be  neglected. 

137.  Add  exactly  16.21,  48.32f 
16,2\   ^16£S\ 

18,32-  =s  48,32-  Since  ^  is  tenths  and  |  is  hundredths^  it  is  neces- 

•  ^  sary  to  reduce  |  to  himdredths. 

64.561 

In  division,  it  is  frequently  possible  to  simplify  by  multiplying 
both  dividend  and  divisor  by  some  number. 

13a  Divide  60  by  2.8^. 

2.8\)60( 

We  multiply  both  terms  by  7,  and  proceed  as 
2,0^0)42.0^  in  Ex.  67. 

21 

139.  Divide  .365f  by  3.8}. 

3.8 g). 365 ^{  We  multiply  both  terms  by  6,  and  proceed  as 

23.2)2.195(  ^^ 


DECIMALS  118 

CIRCULATING  DECIMALS 

If  a  circulating  decimal  is  to  be  subjected  to  any  operation,  it 
is  best  first  to  reduce  to  a  common  fraction. 

To  reduce  a  circulating  decimal  to  a  common  fraction,  for  the 
numerator,  write  the  repetend,  and  for  the  denominator  as  many  9^s 
cw  there  are  figures  in  the  repetend. 

140.  Reduce  .15  and  .207  to  common  fractions  and  illustrate 
the  rule. 

99     33  999      111 


.15  X  100  =  15.1516 .207  x  1000  =  207.207 

.is  X      1=      .1515 .207  X        1=        .207 


.16  X 

99=15 

.207  X    999  =  207 

.16  =  U 

.201 

r  =  H5 

141.   Reduce  .0315  to  a  common  fraction. 

,0315  : 

=  Mf^  = 

'OSh 

lOU 

ss 
100 

104 
3300 

26 
825 

By  the  rule,  . 

0316  =  . 

03U- 

142.   Reduce  1.5  to  a  common 

fraction. 

1.5  = 

.15  X  10 

1.6x100  = 
i.6x     1  = 

161.515 
1.515 

.  .  .  . 

2Q^l^y^20  =  —  1.6x99=150 


99  99 


1.6  =  W  =  UJ 


Find  the  value  of: 

14a   .3 +  .1^  +  1.6  147.  6  4- .6 +  .6 

144.  23.36-16.4  14a  9.16 -.t3 

145.  .25  X  1.6  149.  5.'00'3  X  6 

146.  .625  +  2.7  isa  15.324-1-81 

AMER.    ARITH. —  8 


114  SPECIAL  CASES 

SPECIAL  CASES 

The  decimal  and  the  per  cent  equivalents  of  a  few  common 
1  rivctions  are  used  so  frequently  that  they  should  be  memorized. 

151.  Memorize  Uie  decimal  and  the  per  cent  equivalents  : 

i,  .50,  50%.  J,  .40,  40%  f  .37i,  37i%. 

i,  .33i,  33i%.  f,  .60,  60%.  |,  .62J,  62i%. 

f  .66|,  66}%.  f  .80,  80%.  J,  .STJ,  87J%. 

J,  .25,  25%.  i,  .16},  16|%.  f  .Hi,  11J%. 

},  .75,  75%.  },  .83},  83J%.  ^,  .08},  8}%. 

I,  .20,  20%.  },  .12},  12}%.  ^,  .06},  6J%. 

Thoa:  i,  60  hundredths,  W  per  cent. 

Note.  —  When  the  denominator  of  a  decimal  is  100,  it  may  be  expressed  by  the 
sign  %,  read,  per  cent. 

State  rapidly  the  decimal  equivalents: 

IM.   h  h  h  i-  155.    },  },  },  ^.  15a   i,  },  ^,  f 

IM-   i  t.  »»  h  156.    },  },  J,  }.  159.    f ,  },  ,V,  }. 

!»*•    i  t»  A»  A-         157.    },  f ,  },  tV.  1«X    f ,  J,  f ,  f 

Ex.  152.    .20,  .33J,  .25,  .50. 

State  rapidly  the  per  cent  equivalents: 

161.  h  h  h  h  16*    tV.  h  h  f .  1«7.    f,  h  h  i- 

162.  I,  ^,  },  i.  165.    },  f,  },  tIj.  16a    ^j,  },  f,  f. 

1«3-    i.  h  A»  f  166.    i  I,  },  }.  169.    h  iV.  h  h 

Ex,  161.    76%,  83}o/o,  87i%,  20%. 

Staie  rapidly  the  fractional  equivalents: 

170.  87}%,  661%,  33}%.  173.  37}%,  40%,  8}%. 

171.  16}%,  60%,  25%.  174.   62}%,  16|%,  20%. 

172.  62}%,  20%,  12}%.  175.   66|%,  50%,  11}%. 
Ex.  170.    I,  I,  I 


DECIMALS  115 

In  multiplication  and  division,  it  is  easier  to  use  the  common 
fractions  than  to  use  the  decimal  or  the  per  cent  equivalents. 

Find  mentally  the  value  of: 

176.  24  X  i,  24  X  .16i,  24  x  16|%.  179.  84  x  83^%. 

177.  36  X  i,  36  X  .66|,  36  x  66J%.  180.  76  x  75%. 
17a   64  X  I,  64  X  .37i,  64  x  37^%.  181.  81  x  lli%. 

Find  mentally  the  value  of: 

182.  72^1,  72H-.37i,  72-5-37i%.  185.  49 -^87^%. 

183.  60 -f- 4,  60-^.83J,  60-f-83i%.  186.  65 -5-83i%. 

184.  84  H- J,  84-5-.66J,  48-h66{%.  187.  64 


What  is  the  cost  of: 

188.  48  articles  at  6J^?  8^^?  12J^?  16|^?  33^^?  66|^? 

189.  72  articles  at  87i^?  75^?  83^^?  37^^?  12^^?  25^? 

190.  36  articles  at  $1.33J?  $1.25?  $1.75?  $1.66|?  $2.16J? 
Ex.  190.     $48;  $1.33^  =  9H;  36  X  $1^  =  $48. 

Hoio  many  articles  can  be  bought  for : 

191.  $24@37i^?  75^?  66J^?  40^?  12^^?  80^?   33^^? 

192.  $30@83i^?   62^^?   25^?   37J^?   60^?   66§^?   50^? 

193.  $40  @  $1.25?    $1.33i?    $1.66|?    $2.50?    $3.33J? 
Ex.  193.     32  ;  $  1.26  =  $  | ;  $40  -f-  $  f  =  32. 

194.  What  is  the  cost  of  cloth  per  yard,  when  $  1  is  the  cost 
of  2  yards?  3  yards?  4  yards?  5  yards?  6  yards?  8  yards? 
D  yards?     10  yards?     12  yards?     16  yards? 

195.  How  many  yards  of  cloth  can  be  bought  for  $1  at  50^ 
a  yard?  33^^?  25^?  20^?  16}^?  12^^?  llj^?  8^^?  6J^? 
10^?   9^^? 


116  SPECIAL  CASES 

In  multiplication  and  division,  it  is  easier  to  use  the  common 
fractions  than  to  use  the  equal  (aliquot)  parts  of  100. 

State  the  rule  for: 

196.  Multiplying  by  20.  201.  Dividing  by  50. 

197.  Dividing  by  20.  202.  Multiplying  by  SSJ. 
19a  Multiplying  by  25.  203.  Dividing  by  33^. 
199.  Dividing  by  25.  204.  Multiplying  by  12^. 
20a  Multiplying  by  50.  205.  Dividing  by  12^. 

Ex.  198.    Multiply  by  100  and  divide  by  4 ;   since  25  =  i^ft. 
Ex.  199.    Multiply  by  4  and  divide  by  100 ;   since  25  =  ija. 

Find  the  value  of: 

206.  3683x25.  210.  4686x50.  214.  9678  x  16|. 

207.  3876-5-25.  211.  4686 -i- 50.  215.  9678-f-16|. 
20a  4773  x33J.  212.  7265x20.  216.  6272  x  12J. 
209.  4873  ^33f  213.  7265-5-20.  217.  6272 -5- 12f 

When  the  price  of  articles  is  given  by  the  100  or  1000,  it  is 
easier  to  reduce  the  number  of  articles  to  hundreds  or  thousands. 

21a  How  much  will  13625  bricks  cost  at  $4.75  per  M  (1000)  ? 
Suggestion.— There  are  13.625  M.  at  $4.75  per  M. 

219.  How  much  will  1875  pounds  of  beef  cost  at  $  7.25  per  C 

(100)? 

220.  How  much  will  1975  pounds  of  hay  cost  at  $  6.75  per  ton 
(2000  pounds)  ? 

221.  How  much  will  2146  feet  of  lumber  cost  at  $  22.50  per  M  ? 

222.  How  much  will  875  pounds  of  binder  twine  cost  at  $  6.25 
per  C  ? 


DECIMALS  117 

ANALYSIS 

In  answers  to  problems  involving  U.  S.  money,  the  mills  are 
usually  dropped,  but  5  or  more  are  reckoned  as  an  additional  cent. 

223.  How  much  will  13 J  yards  of  cloth  cost  at  $  1.25  a  yard  ? 
13 J  yards  ? 

Ans.  Iie.sec^  16.6625);  |16.88($  16.876). 

Before  multiplying,  it  is  often  wise  to  determine  the  lowest  order 
required  in  the  product,  and  to  neglect  unnecessary  orders  in  the 
multiplicand. 

224.  In  $  1.23456789  x  52,  what  decimal  orders  may  be  neg- 
lected in  the  multiplicand  ?     Find  the  product  true  to  cents. 

The  product  must  be  true  to  cents,  or  to  two  decimal  places.  It  is  well  to 
find  one  more  decimal  order  than  is  required,  because  the  figure  in  the  last 
order  will  not  always  be  exact.  The  highest  order  in  the  multiplier  is  tens; 
tens  X  ten-thousandths  =  thousandths ;  all  to  the  right  of  ten-thousandths 
may  be  neglected. 

225.  In  $20.123456789x3200,  what  decimal  places  may  be 
neglected  in  the  multiplicand  ?  Find  the  product  true  to  cents ; 
to  mills. 

226.  Find  the  value  of  $(1.06)^*'  x  200,  true  to  cents. 

We  will  find  3  decimal  places  in  the  answer.  The  highest  order  in  the 
multiplier  is  hundreds;  hundreds  x  hundred-thousandths  =  thousandths;  all 
to  the  right  of  hundred-thousandths  may  be  neglected  ;  i.e.,  we  must  retain 
6  decimal  places. 

(1.06)10=  1.06  X  1.06  X  1.00  x  .  .  .  .  where  1.00  is  taken  10  times  as  a 
factor.  1.06  x  1.00,  or  (1.06)«=  1.1230;  1.1230  x  1.1230,  or  (1.06)*  = 
1.20247;  1.26247x1.26247,  or  (1.00)"  =  1.60.383;  1.69383x1.12:16,  or  (l.OO)* 
x(1.06)«,  or  (1.06)W,  =  1.79083,  $1.79083  x  200  =  <>368.17.  The  pupil 
should  perform  these  multiplications. 

227.  How  many  decimal  places  are  there  in  the  exact  value  of 
(1.06)'«? 

22a  If  $1  amounts  to  $(1.06)*  in  5  years,  how  much  will 
$234.63  amount  to  in  the  same  time? 


118  ANALYSIS 

229.  Find  the  cost  of  5|  cords  of  wood  at  $  3.25  a  cord ;  of  7  J 
tons  of  hay  at  $4.75  a  ton ;  of  53}  bushels  of  corn  at  23^  a  bushel. 

230.  The  distance  around  a  wheel  is  3.1416  times  its  diameter. 
How  many  times  will  a  wheel  4  feet  in  diameter  turn  around  in 
going  a  mile  (5280  feet)  ? 

231.  A  wheel  3  feet  in  diameter  turns  286  times  in  going  a 
certain  distance.     What  is  the  distance  ? 

232.  If  a  man's  income  is  $2000  a  year  (365  days),  and  his 
average  expenses  are  $3.68  a  day,  in  how  many  days  will  he  save 
$75?    Prove  the  answer. 

233.  What  is  the  average  expense  per  day  of  a  newspaper  that 
costs  $  1.50  per  year  ? 

234.  At  an  average  expense  of  5^  per  day,  how  much  does  a 
man  spend  for  tobacco  in  20  years  ? 

235.  There  are  2150.42  cubic  inches  in  a  bushel  and  231  cubic 
inches  in  a  gallon.  How  many  bushels  are  there  in  86.378  gal- 
lons? 

236.  When  pork  sells  in  the  market  for  $4.25  per  hundred 
pounds,  how  many  pounds  can  be  bought  for  $  340  ? 

237.  From  a  granary  containing  287  bushels  of  corn,  .36  of  the 
whole  was  sold  at  23^  per  bushel.  How  much  was  received 
from  the  sale  ? 

23a  A  man  worked  221  days  at  the  rate  of  $  1.15  per  day,  and 
took  his  pay  in  wheat  at  62^^  per  bushel.  How  many  bushels 
did  he  receive  ? 

239.  A  farmer  raised  6768  bushels  of  corn  from  160  acres  of 
land,  at  an  average  cost  of  $  6.25  per  acre.  How  much  will  he 
gain  if  he  sells  the  whole  at  18  ^  per  bushel  ? 

240.  On  a  certain  day  the  sales  at  a  lumber  yard  were :  5280  feet 
rough  lumber  at  $14.50  per  M;  3060  feet  dressed  lumber  at 
$17.25  per  M;  4327  feet  yellow  pine  flooring  at  $42.50  per  M; 
224  pounds  lime  at  6\^  per  pound.  To  what  did  the  sales 
amount  ? 


DENOMINATE   NUMBERS 


ENGLISH  SYSTEM 

Substances  may  be  measured  in  various  ways  to  determine  how 
much  they  possess  of  certain  attributes,  such  as  length,  surface, 
volume,  capacity,  weight,  value,  .... 

A  denominate  number  answers  the  question,  How  much  ?  See 
p.  7.     Thus : 

How  much  length?  How  much  surface?  How  much  volume?  How 
much  capacity  ?  How  much  weight  ?  How  much  value  ?  The  answers  ; 
4  feet,  06  square  feet,  64  cubic  feet,  50  quarts,  2  tons,  $50,  are  denominate 
numbers. 

In  naming  units  of  measure,  it  is  thought  best  to  consider 
several  smaller  units  as  forming  a  unit  of  larger  measure ;  several 
of  this  larger  unit  as  forming  a  unit  of  still  larger  measure ;  .  .  .  . 
Thus : 

12  inches  make  1  foot;  3  feet  make  1  yard;  5\  yards  make  1  rod.  Each 
of  these  denominations  is  a  unit  of  measure. 

It  is  difficult  to  conceive  of  a  system  of  measures  more  unscien- 
tific and  more  confusing  than  the  English.  Not  only  do  the  same 
denominations  occur  in  different  tables  ^vith  different  values,  but 
there  are  four  different  tables  for  weight,  two  for  capacity,  and  a 
multitude  of  miscellaneous  units.  The  same  number  of  each 
unit  does  not  make  one  of  the  next  higher;  the  names  of  the 
smaller  and  the  larger  units  in  the  same  table  have  nothing  in 
common ;  the  reduction  from  one  denominatioo  to  another  in  the 
same  or  different  tables,  involves  much  labor.  Compare  with 
metric  systemy  p.  136. 

119 


120  ENGLISH  SYSTEM 

LENGTH 

The  length  of  a  line  is  the  number  of  linear  units  which  it 
contains. 

Thus:  the  line,  A,  contains  the  linear  unit, 
^  j^         If ,  3  times ;  its  length  is  3  units.     If  3f  is  1 

• ' ' '        • '      inch,  ^  is  8  inches ;   if  Jf  is  1  foot,  ^  is  3 

feet ;  .  .  .  . 

The  units  of  length  are :  incfi,  in. ;  foot,  ft ;  yard,  yd. ;  rod,  rd. ; 
miley  mi. ;  chain,  ch. ;  link,  li.  The  surveyors*  chain  is  4  rd.,  or 
66  ft,  or  792  in.,  long,  divided  into  100  links  of  7.92  in.  each ;  the 
engineers*  chain  is  100  ft  long,  divided  into  100  li.  of  1  ft  each. 

ScBTBYOKS^  Long  Measdrx 

7.92  in.  =  1  li. 
26  li.      =  1  rd. 

80  ch.    =  1  mi 


1.  What  instruments  for  measuring  length  have  you  seen? 
What  does  a  carpenter  use  ?  a  tailor  ?  a  dry-goods  clerk  ? 

2.  Draw  a  line  12  units  long.  If  each  unit  is  an  inch,  what 
unit  is  the  whole  line  ?  Draw  a  line  5 J  units  long.  If  each  unit 
is  a  yard,  what  unit  is  the  whole  line  ? 

a  Draw  a  square  6  units  long  and  separate  into  36  equal 
squares.  If  each  linear  unit  is  a  mile,  what  is  the  length  in  rods 
of  one  side  of  each  small  square  ?  of  the  large  square  ? 

4.  In  example  3,  separate  one  of  the  smaller  squares  (a  section) 
into  4  equal  squares.     How  long  is  a  quarter-section  ? 

5.  Draw  a  line  an  inch  long ;  separate  into  2  equal  parts ;  each 

half  into  2  equal  parts ;  each  quarter  into  2 
ABCDSF  '   '  i/   ^^^^^^  P^^^s.      What  is  the  length  of  ABl 
of  ^C?  oiAM"^  oiAE?  oi  AF? 


Long  Measdrjb 

12  in.      =  1  ft 
3  ft.        =1  yd. 

16>  ft.  1  ^  "^ 
320  rd.    =lmi. 

NOTE.- 

3  mi.;  .  . 

—  Other  units  are :  hi 

M 


DENOMINATE  NUMBERS  121 

SURFACE 

The  area  of  a  surface  is  the  number  of  square  units  which  it 

contains. 

Thus:  the  surface,  A^  contains  the  square  unit, 
3f,  6  times  ;  its  area  is  6  square  units.  If  one  side 
of  itf  is  1  in.,  ^  is  6  square  inches ;  if  one  side  of 
Jf  is  1  ft.,  .4  is  6  square  feet ;  .  .  .  . 

The  tables  are  formed  by  squaring  the  units  of  long  measure : 
12  in.  =  1  ft.,  144  sq.  in.  =  1  sq.  ft. ;  3  ft.  =  1  yd.,  9  sq.  ft.  =  1 
sq.  yd.;  5 J  yd.,  or  16^  ft.  =  1  rd. ;  30^  sq.  yd.,  or  272J  sq.  ft.  = 
1  sq.  rd.     Other  units  are :  acre,  A. ;  township,  Tp. 

Square  Measure  Sdrteyors*  Square  Measure 

144  sq.  in.       =  1  sq.  ft  625  sq.  li.  =  1  sq.  rd. 

9  sq.  ft.  =  .1  sq.  yd.  16  sq.  rd.  =  1  sq.  ch. 

30i  sq.  yd.  1  _  .  ,  10  sq.  ch.  =  1  A. 

272^  sq.  ft.  j  ~     ^  640  A.        =1  sq.  mi. 

160  sq.  rd.      =  1  A.  36  sq.  mi.  =  1  Tp. 

Note.  —  The  surveyors'  chain  was  made  4  rd.  long,  because  a  tenth  of  an  acre 
is  16  sq.  rd.,  or  a  square  whose  side  is  4  rd. 

6.  Separate  a  square  3  units  long  into  9  equal  squares.  If  each 
linear  unit  is  a  foot,  what  unit  is  the  large  square  ? 

7.  Draw  a  square  5^  units  long,  making  a  dot  at  the  end  of 
each  unit.  Connect  the  dots  by  lines  parallel  to  the  sides.  How 
many  whole  squares  are  there  ?  How  many  half  squares  are 
there  ?  What  part  of  one  of  the  25  equal  squares  is  the  small 
square  in  one  corner  ?  If  each  linear  unit  is  a  yard,  what  is 
the  area  of  the  large  square  ? 

a  Draw  a  rectangle  4  units  long  and  2  units  wide ;  separate 
into  strips,  and  separate  one  of  the  strips  into  squares.  What  is 
the  relation  between  the  number  of  squares  in  each  strip  and  the 
number  of  linear  units  in  the  length  ?  between  the 
number  of  strips  and  the  number  of  linear  units  in 
the  width  ?    What  is  the  area  of  the  rectangle  ? 


.f=m)  0 


122  ENGLISH   SYSTEM 

VOLUME 

The  \roluine  of  a  solid  is  the  nninber  of  cubic  units  which  it 
contains. 

Thu« :  the  solid,  A,  contains  the  cubic  unit, 
Jf,  3  times  ;  its  volume  is  3  cubic  units. 
If  one  side  of  Jf  is  1  in.,  ^  is  3  cubic  inches ; 
if  1  ft.,  ^  is  3  cubic  feet ; . . . . 

The  tables  are  formed  by  cubing  the  units  in  long  measure : 
12  in.  =  1  ft. ;  1728  cu.  in.  =  1  cu.  ft ;  .  .  .  .  A  pile  of  wood  8  ft 
long,  4  ft.  wide,  and  4  ft  high,  is  called  a  cord,  cd. 

Cubic  Measurb  Wood  Mbasubb 

1728  cu.  in.  =  1  cu.  ft  16  cu.  ft  =  1  cd.  ft 

27  cu.  ft,  =  1  cu.  yd.  8  cd.  ft  =  I  cd. 

Note.  — A  perch  of  ston©  or  masonry  is  164  ft.  long,  1)  ft.  wide,  and  1  ft.  high, 
aoJ  contains  *iu  cu.  ft. 

9.  Draw  a  cube  whose  length  is  3  linear  units,  and  separate 
into  small  cubes.  If  each  linear  unit  is  a  foot,  how 
long  is  one  side  of  the  cube?  What  is  the  area  of 
each  face?  What  is  the  entire  surface  of  the  cube? 
If  each  linear  unit  is  1  yd.,  answer  the  above  questions. 
la  In  example  9,  if  each  linear  unit  is  a  foot,  what  unit  is  the 
large  cube  ?  each  small  cube  ?  What  is  the  volume  of  the  large 
cube  in  cu.  ft  ?  in  cu.  yd.  ?  What  is  the  volume  of  each  small 
cube  in  cu.  ft  ?  How  many  times  does  the  large  cube  contain 
the  small  cube  ? 

U.   Draw  a  prism  4  units  long,  2  units  wide,  and  2  units  high ; 
////yA    separate  into  layers,  and  separate  one  of  the  layers 
into  cubes.     What  is  the  relation  between  the  num- 
ber of  cubes  in  each  layer,  and  the  product  of  the 


i 


linear  units  in  the  width  and  the  height  ?  between  the  number  of 
layers  and  the  number  of  linear  units  in  the  length  ?  "WTiat  is 
the  volume  of  the  prism  ? 


DENOMINATE  NUMBERS  128 

CAPACITY 

The  capacity  of  a  vessel  is  the  number  of  volume  units  which 

it  contains. 

c 
Thus :  C  will  hold  the  measure,  M,  20  times ;  its  capacity  is 
20  volume  units.      U  M  is  &  pint,  the  capacity  of  C  is  20       ^ 
pints ;   if  a  quart,   C  is  20  quarts ;   if  a  gallon,  C  is  20  gal-     ^ 
Ions;  .... 

There  are  two  sets  of  measures,  one  for  liquids,  and  one  for  dry 
commodities.  The  units  are :  gillj  gi. ;  pintj  pt. ;  quart,  qt. ;  gallon, 
gal. ;  barrel,  bbl. ;  bushel,  bu. 

Liquid  Measukb  Dry  Measurb 

4  gi.  =  1  pt.  2  pt.  =  1  qt. 

2  pt.  =  1  qt.  8  qt  =  1  pk. 

4  qt.  =  1  gal.  4  pk.=  1  bu. 

12.  Name  an  article  that  is  sold  by  the  gal.;  by  the  barrel. 
Name  an  article  that  is  sold  by  the  dry  qt. ;  by  the  peck ;  by  the 
bushel. 

13.  Draw  a  rectangle,  11 J  in.  long  and  5f  in.  wide;  draw  the 

fisL^  BEFC.   Cut  out  the  rectangle  id EFZ),  roll  ^Z>  f bji 

to  BC  and  paste  BEFC;   a  cylinder  holding  a    * 

liquid  quart  will  be  formed.  fc ^ 

14.  In  the  same  way  prepare  a  cylinder  holding  a  dry  quart, 
making  the  dimensions  of  the  rectangle  12J  in.  and  5J  in.  Notice 
that  the  dry  quart  is  larger  than  the  liquid  quart. 

15.  Since  a  cube  is  a  volume  unit,  capacity  may  also  be  ex- 
pressed by  cubic  measure.  A  bu.  is  2150.42  cu.  in. ;  a  gal.  is  231 
cu.  in.     How  many  cu.  in.  in  a  dry  qt.  ?  in  a  liquid  qt.  ? 

16.  Capacity  may  be  determined  by  weighing,  and  in  many 
states  the  number  of  pounds  of  the  different  grains  to  be  reckoned 
a  bu.  is  fixed  by  law.  How  many  bu.  of  corn  are  there  in  224  lb., 
if  a  bu.  =  66  lb.  ? 


124  ENGLISH  SYSTEM 

WEIGHT 

For  weight,  there  are  four  sets  of  measures:  troy,  for  the 
precious  metals;  apothecaries',  for  dry  medicines;  apothecaries' 
fluid,  for  liquid  medicines;  avoirdupois,  for  the  coarser  articles 
of  merchandise. 

The  troy  units  are :  grain,  gr. ;  pennyrceighty  pwt. ;  ounce,  oz. ; 
pound,  lb.  The  apothecaries':  grain,  gr. ;  scruple,  3;  dram,  3; 
ounce,  5  ;  pound,  lb;  minim,  ^, -,  pint,  O. ;  gaUon,  Cong.  The 
avoirdupois :  ounce,  oz. ;  pound,  lb. ;  hundredweight,  cwt. ;  ton,  T. 

Tbot  Atoirbcpois 

24  gr.    =lpwt  16  OS.    =  1  lb. 

20  pwt  =  1  o£.  100  lb.  =  1  cwt 

12  oz.     =  1  lb.  20  cwt  =  1  T. 

Apothbcaribs*  Apothecaries*  Fluid 

20gr.  =  13  60nv,=lf3 

33    =13  8f3  =lf5 

8  3     =15  16f5=10. 

12  5   =  1  lb  8  O.    =  1  Cong. 

Note.  —  The  English,  or  long  ton  of  2240  lb.,  is  used  at  the  U.  S.  Custom  House. 

17.  The  grain  is  the  same  for  all  measures ;  5760  gr.  =  1  troy 
or  apothecaries'  pound ;  7000  g^.  =  1  avoirdupois  pound.  How 
many  gr.  make  a  troy  or  apothecaries'  oz.  ?  an  avoirdupois  oz.  ? 

la  60  drops  make  a  teaspoonful;  8  teaspoonfuls  make  an 
ounce.  30  drops  are  what  part  of  a  teaspoonful  ?  How  many 
doses  of  10  drops  to  a  dose,  are  there  in  an  ounce  of  medicine  ? 

19.  A  pint  of  water  weighs  an  avoirdupois  pound  (nearly). 
Why  do  16  oz.  instead  of  12,  make  a  pint  apothecaries'? 

20.  In  writing  prescriptions,  physicians  employ  the  small 
letters  of  the  Roman  notation,  placing  the  symbols  first,  and 
writing  j  for  final  i.     Read:    5ij  3iij  3ij  gr.  xij. 


DENOMINATE  NUMBERS  126 

TIME  AND  ARC 

A  day,  da.,  is  the  time  of  the  revolution  of  the  earth  upon  its 
axis;  a  month,  mo.,  the  time  of  the  revolution  of  the  moon  around 
the  earth;  a  year,  yr.,  the  time  of  the  revolution  of  the  earth 
around  the  sun.  The  other  units  are :  second,  sec. ;  minute,  min. ; 
hour,  hr. ;  tceek,  wk. ;  century,  cen. 

Circular  measure  is  used  in  measuring  arcs  and  angles.  The 
units  are:  second,  ";  minute,  ';  degree,  °',  sign,  S;  circumference,  C. 
The  sign  is  rarely  used. 

TiMB  Circular  Measure 

60  sec.  =  1  min. 

60  min.  =  1  hr. 

24  hr.  =  1  da. 

7  da.  =1  wk. 
62  wk.  1  da.  =  1  yr. 

12  mo.  =  1  yr. 

366  da.  =1  yr. 

366  da.  =  1  leap  year. 

100  yr.  =  1  cen. 

Note.  —  30°  =  1  sign ;  12  signs  =  1  circumference ;  90^  =  1  right  angle. 

21.  How  many  degrees  are  there  in  a  right  angle  ?  in  the  cir- 
cumference of  a  circle  ?    What  is  the  angular  space  about  a  point  ? 

22.  Name  the  months  and  give  the  number  of  days  in  each. 
January  is  the  first  month ;  name  and  give  the  ordinal  numeral 
for  each  month. 

2a  A  year  lacks  674  sec.  of  being  365^  days.  The  fourth  of  a 
day  is  disregarded  for  3  years  and  a  whole  day  is  added  to 
February  every  year  that  is  divisible  by  4.  How  much  of  an 
error  does  this  plan  make  in  400  yr.  ? 

24.  How  is  this  error  of  3  da.  in  400  yr.  corrected  ?  Is  1900 
a  leap  year?  is  2000? 

Ans.  TJte  addition  of  1  day  to  centennial  years  is  omitted  unless 
the  centennial  year  is  divisible  by  400. 


126 


ENGLISH  SYSTEM 
VALUE 


The  units  in  U.  S.  and  Canada  money  are:  mill,  m.  (Latin, 
mille,  1000) ;  cetUf  t  (Latin,  centum,  100) ;  dime,  d.  (Latin,  decern, 
10);  dollar,  $;  eagle^  £. 

The  units  in  English  money  are :  farthing,  far.  or  qr. ;  penny, 
y\ur2A,  j)enc€,  d. ;  ahilling,  s.;  pound,  £  ($4.8665). 

The  French  units  are :  centime,  decime,  fi-anc  (19.3  cents). 

The  German  units  are :  pfennig,  mark  (23.85  cents). 


Ukitbd  Statbs  Mohbt 

10  f    =ld. 

10  d.  =«1 

#10      =1E. 


English  Momst 

4  far.  =  1  d. 
12  d.  =  1  s. 
208.   =£1 


FSBNCH   MOVBT 

10  centimes  =  1  decline 
10  decimee  =  1  franc 


Gbbman  MomtT 
100  pfennigs  =  1  mark 


MoTB.  —  The  English  use  also  croum  (5  shillings) ;  florin  (2  shillings) ;  sover- 
eign {£!). 


The  pupil  should  thoroughly  memorize  the  following  equiva- 
lents: 

12  units  =1  dozen 
12  dozen  =  1  gross 
12  gross  =  1  great  gross 
7000  gr.   =  1  lb.  avoir. 
00  ni       =1  teaspoonf  ul 
48  lb.       =  1  bu.  barley 
56  lb.       =1  bu.  com 
32  lb.       =  1  bu.  oats 
56  lb.       =1  bu.  rye 
60  lb.       =  1  bu.  wheat 
water        60  lb.       =1  bu.  potatoes 


24  sheets 

=  1  quire 

20  quires 

=  1  ream 

480  sheets 

=  1  ream 

231  cu.  in. 

=  1  gal. 

2150.4  cu.  in. 

=  Ibu. 

4  bu.  (approx.) 

=  5  cu.  ft. 

7|  gal.  (approx.  )=  1  cu.  ft 

24  hours 

=  360° 

5760  gr. 

=  1  lb.  troy 

5760  gr. 

=  1  lb.  apoth. 

62|  lb. 

=  weight  1  cu.  f  u 

DENOMINATE  NUMBERS 


127 


EXERCISES 
U.  S.  Money 


How  many : 

25.  m.  make  3  1  ? 

26.  d.  make  $  1  ? 

27.  ^    make  $  1  ? 
2a  ^    make  IE.? 

29.  d.  make  IE.? 

30.  nickels  make  $  1  ? 

31.  quarters  make  $  1  ? 


32.  m.  make  Id.? 

33.  m.  make  IE.? 

34.  m.  make  1  ^  ? 

35.  ^    make  Id.? 

36.  ^   make  IE.? 

37.  halves  make  $  1  ? 

38.  ^  make  1  quarter  ? 


Foreign  Monet 


Hoic  many  : 

39.  far.  make  £1? 

40.  s.  make  £1? 

41.  s.  make  1  crown  ? 

42.  s.  make  1  florin  ? 

43.  far.  make  Is.? 

44.  centimes  make  1  franc  ? 

45.  cents  make  1  franc  ? 


46.  d.  make  Is.? 

47.  d.  make  1  crown ;? 
4a  d.  make  £1? 

49.  far.  make  Id.? 

50.  dollars  make  £1? 

51.  pfennigs  make  1  mark  ? 

52.  cents  make  1  mark  ? 


Troy  and  Avoirdupois  Weights 


How  many : 

53.  gr.  make  1  lb.  (troy)  ? 

54.  gr.  make  1  oz.  (troy)  ? 

55.  pwt.  make  1  lb.  ? 

56.  pwt.  make  1  oz.  ? 

57.  oz.  make  1  lb.  (troy)  ? 
5a  gr.  make  1  lb.  (avoir.)? 
59.  gr.  make  1  oz.  (avoir.)? 


60.  gr.  make  1  lb  (apoth.)  ? 

61.  oz.  make  1  lb  (apoth.)  ? 

62.  cwt.  make  IT.? 

63.  lb.  make  IT.? 

64.  gr.  make  1  T.  ? 

65.  lb.  make  1  long  T.  ? 

66.  oz.  make  1  lb.  (avoir.)? 


128 


ENGLISH  SYSTEM 


Apothecaries*  Wbioht 


How  many : 

67.  m,  make  1  f  3  ? 

ea  f3  make  1  f5? 

69.  f  3  make  10.? 

70.  f  5  make  1  Cong.? 
71-  f  3  make  1  Cong.  ? 
72.  drops  make  1  f  3  ? 

7a  drops  make  1  teaspoonful  ? 


74.  gr.  make  1  !b  ? 

75.  gr.  make  15? 

76.  3  make  1  !b  ? 

77.  3    make  1  lb  ? 
7a    3    make  15? 

79.   drops  make  1  f  5  ? 

aa  tcaspoonfulsmakelf5? 


How  many  : 

8L  in.  make  1  yd.  ? 
82.  in.  make  1  rd.  ? 
aa   ft.  make  1  rd.  ? 

84.  ft.  make  1  mi.  ? 

85.  rd.  make  1  mi.  ? 
8a  in.  make  1  hand  ? 
87.  ft.  make  1  fathom  ? 


Loxo  Mbasckb 


88   li.    make  1  eh.  ? 

89.  li.    make  1  rd.  ? 

90.  ch.  make  1  rd.  ? 

91.  ch.  make  1  mi.  ? 

92.  in.   make  1  li.  ? 

9a   rd.  make  1  furlong  ? 
94.  mi.  make  1  league  ? 


How  many 

'.* 

Squabb  Measure 

95. 

sq. 

in. 

make  1  sq. 

ft.? 

102. 

96. 

sq. 

in. 

make  1  sq. 

yd.? 

103. 

97. 

sq. 

ft. 

make  1  sq. 

rd.? 

104. 

9a 

sq. 

rd. 

make  1  A. 

? 

105. 

99. 

sq. 

ft 

make  1  sq. 

yd.? 

loa 

100. 

A. 

make  1  section 

9 

107. 

101. 

A. 

make  1  quarter  section  ? 

108. 

sq.  ch.  make  1  A.  ? 
sq.  li.  make  1  sq.  ch.  ? 
sq.  li.  make  1  sq.  rd.  ? 
A.  make  1  sq.  mi.  ? 
sq.  mi.  make  1  Tp.  ? 
A.  make  1  half  section  ? 
sections  make  1  Tp. 


DENOMINATE  NUMBERS 


129 


Cubic  and  Wood  Measures 


How  many : 

109l  cu.  in.  make  1  cu.  ft,  ? 

no.  cu.  in.  make  1  cu.  yd.  ? 

111,  cu.  ft.  make  1  cu.  yd.  ? 

112.  cu.  in.  make  1  cd.  ft.  ? 
ua  cu.  in.  make  3  cu.  ft.  ? 

114.  cu.  ft.  make  1  j^erch  ? 

115.  cu.  in.  make  1  perch  ? 


116.  cu.  ft.  make  1  cd.  ? 

117.  cu.  ft.  make  1  cd.  ft.  ? 
lia  cu.  ft.  make  3  cu.  yd.  ? 
119.  cu.  ft.  make  2  cu.  yd.  ? 
12a  cu.  ft.  make  3  cd.  ? 

121.  cd.  ft.  make  4  cd.  ? 

122.  cu.  ft.  make  4  cd.  ? 


Liquid  and  Dry  Measures 


How  many : 

123.  pt.  make  1  qt.  (dry)  ? 

124L  qt.  make  1  pk.  ? 

125.  pt.  make  1  bu.  ? 

126.  pt.  make  1  pk.  ? 

127.  pk.  make  1  bu.  ? 
12a  lb.  in  1  bu.  corn  ? 
129.  lb.  in  1  bu.  oats  ? 


130.  pt.  make  1  qt.  (liquid)  ? 

131.  qt.  make  1  gal.  ? 

132.  qt.  make  1  bbl.  of  31|  gal.  ? 

133.  pt.  make  1  bbl.  of  45  gal.  ? 

134.  qt.  make  1  bbl.  of  40  gal.  ? 

135.  lb.  in  1  bu.  wheat  ? 

136.  lb.  in  1  bu.  potatoes  ? 


Miscellaneous 


How  many  : 

137.  da.  make  1  leap  year  ? 

13a  degrees  make  1  C.  ? 

139.  cu.  in.  make  1  gal.  ? 

140.  cu.  in.  make  1  bu.  ? 

141.  degrees  make  IS.? 

142.  da.  in  year  19()0  ? 
14a  da.  in  year  1004  ? 

AMEK.    ARITH.  — 9 


144.  units  make  1  dozen  ? 

145.  units  make  1  score  ? 

146.  sheets  make  1  quire  ? 

147.  sheets  make  1  ream  ? 
14a  units  make  1  gross  ? 

149.  da.  in  year  2000? 

150.  da.  in  year  2004  ? 


130 


ENGLISH  SYSTEM 


TO  LOWER  DENOMINATIONS 
151.  Reduce  2  T.  4  cwt.  3  lb.  6  oz.  to  oz. 


T.       cwt. 

lb. 

oz. 

2     4 

3 

6 

JO 

44  cwt. 

100 

4403  lb. 

16 

1  T.  ia  20  cwt. ;  2  T.,  2  times  20  cwt.,  or, 
with  4  cwt,  44  cwt. 

1  cwt.  is  100  lb.  ;  44  cwt,  44  times  100  lb., 
or,  with  3  lb.,  4403  lb.,  etc. 

Ans.  70,454  oz. 


70454  OZ. 

152.  Reduce  .875  bu.  to  integers  of  lower  denominations. 

.875  bu. 

.  1  bu.  is  4  pk.;  .875  bu.,  .875  times  4  pk.,  or 

±  3.5  pk. 

3.500  pk.  1  pk.  is  8  qt ;  .5  pk.,  .5  times  8  qt ,  or  4  qt. 

3  Ans.  3  pk.  4  qt 
4.0      qt, 

15a  Reduce  |  lb  apothecaries'  to  integers  of  lower  denomi- 
nations. 


|lb  =  |x  7^,  or  e|s. 


^S=|x«,or5i3. 


S 


in)isl2  5;^n)i8f  times  12  5 ,  or  6|  5 . 
15  is  83;  §5  is  f  times  8 3,  or  5^-  1  3  is 
3  3;  i  3  is  i  times  3  3,  or  1  3. 

Ans.  055313. 


Bedtice : 

154.  3  bu.  2  pk.  1  pt.  to  pt.       156.  5iij,  3  vij  to  drops.  Seep.  124. 

155.  £8  3s.  5d.  2  far.  to  far.    157.  1  lb  8  5  7  3  2  3  19  gr.  to  gr. 

Reduce  to  integers  of  lower  denominations  : 
15a  .628  T.  162.  .125  yr.  166.  £  .1416.  170.  .7854  lb. 

159.  .345  mi.        163.  .385  A.  167.  .3285  bu.        171.  .2825  C. 


160.  \\  A. 

161.  |hr. 


164. 


TT 


cu.  yd. 


16a  I 


mi. 
165.  I  lb.  troy.      169.  ^^  T. 


172.  I  gal. 

173.  I  Tp. 


DENOMINATE  NUMBERS  131 

TO  HIGHER  DENOMINATIONS 

174.  Reduce  2736  oz.  avoirdupois   to   integers  of  higher  de- 
nominations. 

16)2736  oz.  10  oz.  are  1  lb.  ;  2736  oz.,  as  many  lb.  as  16 

100)171  lb.  is  contained  times  in  2730,  or  171  lb.  ;  etc. 

1  cwt.  71  lb.  ^ns.  1  cwt.  71  lb. 

175.  Reduce  4  cwt.  3  lb.  8  oz.  to  the  fraction  of  a  T. 
Soz.  =  ^,or'-lb.  ^' 


16'      ' 


4  cwt.  3  lb.  8  oz.  =   6456  oz. 


9I  Jh       '^  ^inn  r.^     '^    ^n*  1  '^'  =  32000  oz. 

3i  lb.  =  -^  m  or  —  cwt.  ^^^  ^^^.^^        ^  ^^^,^  ^, 

^  R07  R07  ~  ^^^  ^' 

176.  Reduce  7  5  3  3  2  3  to  the  decimal  of  a  ft). 
s)2'Sy 

R\  ^  66fi4-  X  3  3  are  1  3 ;  2  3,  as  many  3  as  3  is  con- 

^^  i  tained  times  in  2,  or,  with  3  3 ,  3.666+  3  ;  etc. 

12)7451-^1  ^n,..621+lb. 

.6^i+lb 

Reduce  to  integers  of  higher  denominations: 

177.  8269  gi.  181.  32844  s. 

178.  .'^S46  gr.  troy.  182.  46381  in. 

179.  8637  gr.  ai)othecaries'.         183.  47351  sq.  in. 

180.  6732  in.  surveyors'.  184.  48394  cu.  in. 

liedme  to  common  fractions  of  the  highest  denomination: 

185.  8  5  7  3  1  3  10  gr.  18a  5  ch.  60  li.  3.96  in. 

186.  1  A.  9  sq.  ch.  2000  sq.  li.     189.   2  cd.  7  cd.  ft.  12  cu.  ft. 

187.  1  lb.  9  oz.  2  pwt.  12  gr.       190.  2  T.  15  cwt.  60  lb.  4  oz. 

Reduce  to  decimals  of  the  highest  denomination: 

191.  3  gill.  2  qt.  1  pt.  194.   7.16  T.  6  c\vt.  5  oz. 

192.  5  bu.  3  ])k.  7  pt  195.  40°  20'  15". 

193.  J^6  8s.  6d.  196.  5  yr.  4  mo.  3  wk.  5  da. 


132 


ENGLISH  SYSTEM 


FROM  TABLE  TO  TABLE 
197.   Reduce  288  lb.  avoirdupois  to  lb.  troy. 


288  X  7000 
5760 


=  360 


7000  gr.  =  1  lb.  avoirdupois. 
6760  gr.  =  1  lb.  troy. 


19a  Reduce  87  lb  6  S  apothecaries'  to  lb.  avoirdupois. 


87.5  X  5760 
7000 


=  72 


7000  gr.  =  1  lb.  avoirdupois. 
6760  gr.  =  1  lb.  apothecaries*. 


199.  Reduce  10  bbl.  to  bu.,  approximately. 


J0x^x^xi=SS.6 
2       15     5 


31)  gal.  =  1  bbl. ;  7J  gal.  =  1  cu.  ft. ;  6  cu. 
ft.  =  4  bu. 


200.  Reduce  10  bbl.  to  bu.,  true  to  1  decimal  place. 

m  ..  ^S  ^    231    _ »»  p  31J  gal.  =  1   bbl.;    231  cu.   in.  =  1  gal.; 

1     21504  2150.4  cu.  in.  =  1  bu. 

20L  Reduce  11°  13'  47"  to  time. 

uiin.      sec. 

ii°  =  44  360°  =  24  hr. 

13'  =  52  1°  =  ^  or  T^s  hr.,  or  4  min. 

^7"  =  3*  1'  =  A  Of  1*5  ™i°'  ^^ ^  ^^' 


u 

U    55j-^ 

202.  Reduce  6  hr.  3  min.  to  arc. 
6  hr.    =  90° 
3  7iun.  =  45' 

90°    45' 

Reduce  and  explain  : 

203.  630  lb.  avoir,  to  apoth. 

204.  6  lb.  9  oz.  troy  to  avoir. 

205.  30  cu.  ft.  to  bu.  approx. 

206.  10  bu.  to  bbl.  approx. 


1"  =  «V  or  T»5  sec. 


24  hr.   =360°. 

1  hr.     =  W  or  15°- 

1  min.  =  l^  or  i°,  or  16'. 


207.  10  bu.  to  bbl.,  to  1  dec.  place 

208.  5  bu.  to  gal.  approximately. 

209.  5°  5'  5"  to  time ;  6°  6'  to  time 

210.  5  hr.  5  min.  5  sec.  to  arc. 


DENOMINATE  NUMBERS 


183 


4 

4 

2 

7 

3 

5 

1 

8 

8 

^\ 

1 

S 

1 

6 

ADDITION 
211.   Add :  4  rd.  4  yd.  2  ft.  7  in.,  3  rd.  6  yd.  1  ft.  8  in. 

nl.        v«l.        ft.        in. 


The  sum  of  the  in.  is  15  in.,  or  1  ft.  3  in.  ; 
we  write  3  in  in.  column  and  carry  1  to  ft. 
column. 

■*  write  1  in  ft.  column  and  carry  1  to  yd.  column. 

Since  i  yd.  =  1  ft.  6  in.,      Se.e  p.  20. 
to  avoid   fractions  in  the 
answer,   we  may  proceed 
as  above. 

212.  To  Dec.  5,  1883,  add  165  da. 

Dec.  17 Oy      Mar.  80, 

r«^    i9n        A.r^    I  a  I^^c.  5  +  166  =  Dec.  170.      Since  there  are 

Jan.  139,     Apr.  49,        g^  ^^  ^^  ^^^    ^^^  17^  ^  j^^  139 ;  ...  . 

Feb.  108,      May  19. 

213.  To  Apr.  30,  add  3  mo.  3  da. 
Jtdy  30,   July  33,   Aug.  2. 

214.  Add:  5  T.  4  cwt.  16  lb.  5  oz.,  16  T.  17  cwt.  13  oz.  75  lb., 
3  T.  7  cwt.  12  oz. 

215.  Add :  5  rd.  3  yd.  2  ft.  3  in.,  13  rd.  6  yd.  2  ft.  9  in.,  7  rd. 
2  yd.  5  ft. 

216.  Add:  3  lb.  3  oz.  7  pwt.  22  gr.,  16  lb.  11  oz.  19  pwt.  3  gr., 
7  lb.  14  gr. 

217.  Add:   20  yr.  5  mo.  6  da.  4  hr.,  30  yr.  8  mo.  7  da.  16  hr., 
9  mo.  7  da.  7  hr. 

2ia   Add :  ^  mi.  and  J  rd. ;  first,  reduce  each  fraction  to  in- 
tegers of  lower  denominations. 

219.  Add  J  mi.  and  |  rd. ;  first,  reduce  each  fraction  to  the  deci- 
mal of  a  rod. 

220.  Add  f  lb.  and  f  pwt;  first,  reduce  each  fraction  to  grains. 

221.  To  July  23,  add  93  da.     To  Aug.  5,  add  100  da. 

222.  To  July  23,  add  3  mo.  3  da.     To  Mar.  20,  add  200  da. 


Apr.  .30+3  mo.  =  July  30.    July 
30  +  3  da.  =  July  33,  or  Aug.  2. 


134  ENGLISH  SYSTEM 

STJBTRACTION 

22a  Subtract  20  gal.  2  qt  1  pt  from  30  gal. 

9Q     Q     Q  ^  P^  ^^^^  ^  P^  ^®  cannot  take  ;  we  add 

2  pt  to  0  pt.,  making  2  pt,  and  1  qt.  to  2  qt, 
^0^1  making  3  qt    1  pt  from  2  pt  leaves  1  pt. ;  we 

g     2     1  write  1  in  pt  column.    See  p.  SO. 

224.  Find  the  exact  number  of  days  from  July  20,  1897,  to 
Nov.  19, 1897. 

Jly  July       Sly  Oct, 

*/    Aun         1Q   V/M»  '"  ^^^^  ^^^  are  11  da.  left ;  in  Aug.,  31 ; 

31,  Aug.     jy,  jsov.      jj^  g^       ^    ^^ 

SOj  Sejit.     122y  Am. 

NoTB.  —  This  method  is  used  in  Equation  of  Payments.    500  p.  231. 

225.  Find  the  time  from  July  20,  1897,  to  Nov.  19, 1897. 

1897    11     19 

1897       7    20  ^**^*  '^'  ^®^^'  ^  ^^^  1897th  yr.,  the  11  mo., 

. and  the  19th  da. 

S    29 

NoTK. — This  is  the  method  commonly  used  in  Partial  Pa3rment8.    See  p.  223. 

226.  Subtract  5  lb.  6  oz.  3  pwl  9  gr.  from  8  lb.  4  gr. ;  4  mi. 
100  rd.  4  yd.  from  5  mi. 

227.  Subtract  6  rd.  4  yd.  6  in.  from  16  rd.  1  ft  1  in. ;  3  A.  142 
sq.  rd.  from  5  A. 

22a  From  ^  da.,  subtract  4J  hr.  Give  the  answer  (a)  as  a 
common  fraction  of  a  dayj  (b)  as  a  decimal  of  a  day;  (c)  in 
hr.,  min.,  and  sec. 

229.  Find  the  exact  number  of  days  from  Sept.  3,  1899,  to 
Apr.  4,  1900. 

23a  Find  the  time  from  Sept.  3,  1899,  to  Apr.  4,  1900,  as 
practiced  in  partial  payments. 

231.  How  many  days'  difference  by  the  methods  in  examples 
229  and  230  ? 


DENOMINATE  NUMBERS  135 

MULTIPLICATION   AND   DIVISION 
232.   Multiply  2  T.  3  cwt.  46  lb.  by  8. 

T.       cwt.      lb. 

IS      S      46  8  times  46  lb.  are  368  lb.,  or  3  cwt.  and 

»  68  lb. ;    we  write  68^  in  lb.  column,  and 

carry  3  cwt.    See  p.  S8. 


17      7      68 


23a  Divide  17  T.  7  cwt.  ^  lb.  by  8. 

17  T  -T-  8  =  2  T.  and  1  T.  remaining ;  we 
T-      cwt.     lb.  ^rite  2  in  T.  column. 

8)  17      7      68  1  T.  and  7  cwt.  =  27  cwt. ;  27  cwt.  ^ 

2      S      4.6  8  =  3  cwt.  and  3  cwt.  remaining ;    etc. 

Seep.  51. 

234.  Divide  19  rd.  6  yd.  10  in.  by  2  rd.  2  yd.  2  ft.  2  in. 

19  rd.  5  yd.  10  in.  =3952  in. 

a  ^  m     ,^^^.  ,o/«  We  reduce  both  dividend  and  divisor 

2Td.2  yd.  2  ft.  2  in.  =  m  m.  ^  ^^^  ^^^^  denomination. 

494)3952  {8 

Multiply  : 

235.  10°  36'  48"  by  8.  23a  6  rd.  4  yd.  2  ft.  3  in.  by  6. 

236.  2  bu.  3  pk.  4  qt.  by  9.  239.  4  da.  8  hr.  25  min.  40  sec.  by  7. 

237.  3  gal.  1  qt.  1  pt.  by  5.  240.  4  sq.  yd.  2  sq.  ft.  56  sq.  in.  by  4. 

Divide : 

241.  3  da.  6  hr.  2  min.  by  4.  245.  6  yr.  2  mo.  2  da.  by  12. 

24i  3  mi.  6  ft.  10  in.  by  6.  246.  20  bbl.  3  gal.  by  3  qt.  1  pt. 

243.  15  T.  16  cwt.  4  lb.  by  9.  247.  3  rd.  6  yd.  2  ft.  by  6  yd.  1  ft 

244.  2  bu.  3  pk.  by  5  pk.  1  qt.  24a  £6  63.  6d.  by  3d.  3  far. 


136  METRIC  SYSTEM 

METRIC   SYSTEM  — LAWS 

The  principal  unit  of  each  table  is  derived  ihmh  the  meter. 
The  other  units  are  formed  by  prefixing  to  the  principal  unit  the 
Latin  sub-multiples:  w///i,  1000;  centij  100;  decif  10,  and  the 
Greek  multiples:  deca^  10;  hectOy  100;  kilo,  1000;  myriVi,  10,000. 
i'lie  abbreviation  of  each  sub-multiple  is  the  small  form  of  its  first 
letter ;  of  each  multiple,  the  capital  form. 

SUB-MCLTIPLSt  AHD  MULTIPLM 

10  milli  uniU  =  1  centi  unit  10  deca  unUs  =  1  hecto  unit 

10  centi  units  =  1  deci  unit  10  hecto  units  =  1  kilo  unit 

10  deci  units  =  1  unit  10  kilo  units    =  1  myria  unit 

10  unitfl  =  1  deca  unit 

249.  Substitute  meter  for  unit  in  the  table  above,  and  write  the 
new  table  thus  formed ;  compare  with  long  measure,  p.  137. 

25a  In  the  same  way,  substitute  are,  and  compare  with  the 
table  of  land  measure,  p.  138. 

251.  Substitute  Mere,  liter,  gram,  and  compare  with  tables  for 
wood  measure,  capacity,  weight,  pp.  139,  140, 141. 

252.  Give  the  sub-multiples  for  100,  10,  1000.  State  the  mean- 
ing of  deci,  milli,  centi.     Read  m,  d,  c. 

25a  Give  the  multiples  for  100,  10,  10000,  1000.  State  the 
meaning  of  deca,  kilo,  hecto,  myria.     Read  K,  M,  D,  H. 

254.  How  many  milli  make  1  unit  ?  How  many  centi  ?  How 
many  deci  ? 

255.  How  many  units  make  1  deca?  1  hecto?  1  kilo?  1 
myria  ? 

256.  How  many  m  make  1  M  ?     How  many  c  make  1  H  ? 

Ans.  10,000,000  m  =  1  M  ;  m  =  1000 ;  M,  10,000.  Since  one  is  a  sub- 
multiple  and  the  other  a  multiple,  we  multiply  1000  by  10,000. 

257.  How  many  D  make  1  M  ?     How  many  m  make  Id? 
Ans.    1000  I)  =  1  M ;  D  =  10;  M,  10,000.     Since  both  are  multiples,  we 

divide  10,000  by  10. 


DENOMINATE  NUMBERS  137 


LENGTH 


The  principal  iiiiit  of  long  measure  is  the  meters  ra.  It  is 
designed  to  be  1  ten-millionth  of  the  distance  from  the  equator 
to  the  i)ole;  its  equivalent  is  39.37  in.  The  other  units  are 
formed  by  prefixing  to  the  principal  unit  the  sub-multiples 
and  multiples,  as  explained  on  p.  136.  Approximately,  1  m  = 
1.1  yd.;   1  Km  =  f  mi. 

Long  Measure 

10  mm  =  1  cm  10  Dm  =  1  Hm 

10  cm   =1  dm  10  Hm  =  1  Km 

10  dm  =  1  m  10  Km  =  1  Mm 

10  m  =  1  Dm 

25a  State  the  principal  unit  of  long  measure ;  its  abbreviation ; 
how  obtained;  its  English  equivalent;  approximate  equivalents 
for  1  m,  1  Km. 

259.  Explain  how  the  table  of  long  measure  is  made  up  from 
the  table  of  sub-multiples  and  multiples.     See  Ex.  249. 

260.  To  illustrate  long  measure,  prepare  a  strip  of  paper  39|  in. 
long ;  divide  it  into  10  equal  parts  (dm)  ;  divide  each  dm  into  10 
equal  parts  (cm) ;  divide  the  first  cm  into  10  equal  parts  (mm). 

261.  To  illustrate  long  measure,  place  5  nickels  (5-cent  pieces) 
in  a  row;  the  row  will  be  nearly  1  decimeter  long,  since  the 
diameter  of  each  nickel  is  nearly  2  centimeters.  Place  them  in  a 
pile ;  the  pile  will  be  nearly  1  centimeter  high,  since  each  nickel 
is  nearly  2  millimeters  thick. 

262.  How  tall  are  you  ?  How  long  is  your  arm  ?  What  is  the 
length  of  your  forefinger?  What  is  tlie  thickness  of  your  tlmmb 
nail  ?     What  is  the  width  of  your  thumb  nail  ? 

263.  Prove  that  1  Km  =  f  mi.  (nearly).  Equivalents ;  1  ni  = 
39.37  in.,  1  mi.  =  6280  ft. 

264.  What  is  the  distance  around  the  earth  in  meters?  in 
inches  ?  in  miles  ? 


138  METRIC  SYSTEM 

SURFACE 

Square  measure  is  formed  by  squaring  long  measure.  Thus: 
10  mm  =  1  cm  ;  100  sq  mm  =  1  sq  cm ;  .  .  .  . 

The  principal  unit  of  land  measure  is  the  are,  a ;  it  is  a  square, 
10  m  by  10  m.     Approximately,  1  are  =  ^  of  an  acre. 

SouARB  Mjcasusb  Laitd  Mbasuri 

100  sq  mm  =  lBqcm  10ma=lca 

100  sq  cm   =  1  sq  dm  10  ca  =  1  da 

100sqdm=lsqm  10da=la 

100  sq  m     =  1  sq  Dm  10  a    =  1  Da 

100  sq  Dm  =  1  sq  Urn  10  Da  =  1  Ha 

100  sq  Hm  =  1  sq  Km  10  Ha  =  1  Ka 

100  sq  Km  =  1  sq  Mm  10  Ka  =  1  Ma 

NoTB.  —The  final  vowel  of  each  sab-multiple  and  multiple  is  dropped  before 

art :  millare,  centare ;  not  milliare,  centiare, .... 

The  table  of  land  measure  is  often  given :  100ca  =  la;  100  a  =  l  Ha. 


State  the  principal  unit  of  land  measure ;  its  abbreviation; 
how  obtained ;  its  approximate  equivalent 

266.  Read:   50.23  sq  mm;    238.6  sq  Mm;   560  a;  378.6  Ha 
50.23  sq  Dm. 

267.  How  many  sq  mm  make  1  sq  Km  ?  How  many  sq  Dm 
make  1  sq  Mm  ? 

26a  Is  the  multiple  in  land  measure  10  or  100  ?  In  the  abbre- 
viated form,  why  does  it  appear  to  be  100  ? 

269.  How  could  you  illustrate  land  measure  on  the  school 
grounds  ? 

270.  How  much  land  would  you  like  for  a  flower  garden? 
What  is  the  area  of  this  floor  ?  How  much  land  is  needed  for  a 
good  farm  ? 

271.  Prove  that  1  are  is  nearly  ^^  of  an  acre.  Equivalents: 
1  a  =  10  m  X  10  m ;  1  m  =  39.37  in. 

272.  An  emigrant  exchanges  his  farm  of  4  hectares,  at  12  francs 
an  are,  for  land  in  America  at  $  5  an  acre.  How  many  acres  does 
he  secure  ? 


DENOMINATE  NUMBERS  139 

VOLUME 

Cubic  measure  is  formed  by  cubing  long  measure.  Thus: 
10  mm  =  1  cm  ;  1000  cu  mm  =  1  cu  cm ;  .... 

The  principal  unit  of  wood  measure  is  the  sterCf  s ;  it  is  a  cube, 
Imxlmxlm.     Approximately,  1  stere  =  J  of  a  cord. 

Cubic  Measure  Wood  Measure 

1000  cu  mm  =  1  cu  cm  10  ms  =  1  cs 

1000  cu  cm   =  1  cu  dm  10  cs  =  1  ds 

1000  cu  dm  =  1  cu  m  10  ds  =  1  s 

1000  cu  m     =  1  cu  Dm  10  s     =  1  Ds 

1000  cu  Dm  =  1  cu  Hm  10  Ds  =  1  Hs 

1000  cu  Hm  =  1  cu  Km  10  Hs  =  1  Ks 

1000  cu  Km  =  1  cu  Mm  10  Ks  =  1  Ms 

Note.  —The  table  of  wood  measure  is  often  given,  10  ds  =  1  s,  the  other  units 
being  omitted. 

273.  Explain  how  the  table  of  cubic  measure  is  made  up  from 
the  table  of  long  measure. 

274.  Explain  how  the  table  of  wood  measure  is  made  up  from 
the  table  of  sub-multiples  and  multiples.     See  Ex.  251. 

275.  State  the  principal  unit  of  wood  measure,  its  abbreviation ; 
how  obtained  ;  its  approximate  equivalent. 

276.  How  many  cu  dm  make  1  cu  Hm  ?  cu  Dm  make  1  cu  Mm  ? 

277.  How  could  you  illustrate  a  stere  on  the  school  grounds? 
27a   What  is  the  contents  of  this  room  in  cu  m  ?     How  much 

earth  will  make  a  gootl  load  for  two  horses?     Approximately, 
how  many  steres  are  there  in  a  cord  of  wood  ? 

279.  Prove  that  1  stere  is  nearly  J  of  a  cord.  Equivalents: 
1  stere  =  lmxlmxlm;  lm  =  39.37  in. 

280.  Which  is  the  cheaper,  to  buy  wood  at  $  3  a  cord  or  at  $  1 
a  stere  ? 

281.  What  is  the  cost  of  excavating  40  cu  Dm  of  earth  at  12^  a 
cu  m? 


140  METRIC  SYSTEM 

CAPACITY 

The  principal  unit  of  capacity  is  the  liters  1 ;  it  is  a  cube,  1  dm 
by  1  dm  by  1  dm ;  its  equivalent  is  .908  qt.  dry,  or  1.05  qt  liquid. 
Approximately,  1  1  =  1  qt 

Table  of  Capacity 

10  ml  =  1  el  10  Dl  =  1  HI 

10  cl   =1  dl  10  HI  =  1  Kl 

lOdl  =  1  I  lOKl^  1  Ml 

10  1  =  1  Dl 

282.  State  the  principal  unit  of  capacity;  its  abbreviation ;  how 
obtained ;  its  exact  equivalent ;  its  approximate  equivalent. 

283.  How  much  milk  do  you  need  for  a  cup  of  coffee  ?  How 
much  milk  per  day  would  you  use  for  a  family  of  six?  How 
much  will  a  teacup  hold  ?    How  much  will  a  tablespoon  hold  ? 

284.  How  many  liters  in  23.645  cu  m  ?  How  many  cu  m  in 
385,623  1  ? 

285.  Prove  that  1  liter  is  .908  qt  dry.  Equivalents  :  1  liter  = 
1  dm  X  1  dm  X  1  dm  ;  1  m  =  39.37  in ;  1  bu.  =  2150.4  cu.  in. 

286.  Prove  that  1  liter  is  1.05  qt  liquid.  Equivalents :  1  liter  = 
Idmxldmxldm;  lm  =  39.37  in. ;  1  gal.  =  231  cu.  in. 

287.  Counting  a  liter  as  a  quart,  what  is  the  capacity  in  cu  m 
of  a  bin  that  will  hold  256  bu.  ? 

28a  If  the  bin  in  Ex.  287  is  filled  with  wheat,  what  is  the 
value  of  the  wheat  at  22^  a  Dl  ? 

289.  To  illustrate  measures  of  capacity,  draw  a  rectangle,  as 
ABCDy  whose  length  is  29.16  cm,  and  breadth,  14.78  cm. 
Draw  the   flap   BEFC.     Cut  out  AEFDy  roll  over  AD  to  EC 

f! ^      and   paste  BEFC;   a  cylinder  holding  a  liter 

will  be  formed.     Compare  with  the  liquid  and 


CF      dry  quarts.     See  Exs.  I4  and  15. 


DENOMINATE   NUMBERS  141 

WEIGHT 

The  principal  unit  of  weight  is  the  gram,  g ;  it  is  the  weight  of 
a  cii  cm  of  pure  water  at  its  maximum  density ;  its  exact  equivsr 
lent  is  15.432  gr.    Approximately,  1  Kg.  =  2^  lb.  avoirdupois. 

Table  of  Weight 

10  mg  =  1  eg  10  Dg  =  1  Hg 

10  eg  =  1  dg  10  Hg  =  1  Kg 

10  dg  =  1  g  10  Kg  =  1  Mg 

10  g  =  1  Dg 

Note.  —  There  are  two  additiunal  uuits :  10  myriagrams  =  1  quintal,  Q ; 
10  quiutals  =  1  tonntau,  T. 

290.  State  the  principal  unit  of  weight ;  its  abbreviation ;  how 
obtained ;  its  exact  equivalent ;  the  approximate  equivalent  of  1 
Kg  in  avoirdupois  pounds. 

291.  How  many  dg  in  3  Hg  ?  How  many  eg  in  63  Dg  ?  How 
many  mg  in  46  Kg  ? 

292.  How  many  grams  in  1273.14  gr.  ?  How  many  Hg?  How 
many  grains  in  9  Dg  8  g  ? 

293.  To  illustrate  weights,  procure  a  piece  of  tin  foil  that  weighs 
as  much  as  a  nickel  (5-cent  piece);  cut  the  foil  into  five  equal 
parts.  Each  part  will  weigh  a  gram,  since  a  nickel  weighs  5 
grams.    Procure  a  stone  that  weighs  2  lb.  3  oz. ;  it  will  weigh  1  Kg. 

294.  What  is  your  weight  ?  How  much  beefsteak  would  you 
buy  for  breakfast  for  three  ?  What  is  the  weight  of  a  nickel  ? 
How  heavy  a  letter  of  the  first  class  will  go  for  2^  ?  What  does 
an  average  horse  weigh  ? 

295.  Prove  that  1  cu  m  of  water  weighs  nearly  1  long  ton. 

296.  Prove  that  1  Kg  is  nearly  2^  lb.  av. ;  nearly  2|  lb.  troy. 

297.  Prove  that  the  weight  of  1  gram  is  15.432  gr.  Relations : 
1  g  =  weight  of  1  cu  cm  of  water;  1  m  =  39.37  in.  j  1  cu.  ft.  water 
weighs  62J  lb. 


142 


METRIC   SYSTEM 
APPROXIMATIONS  —  MENTAL  EXERCISES 


The  approximate  equivalents  given  in  the  several  tables  should 
be  memorized,  and  should  be  used  except  when  exact  results  are 
required. 


Reduce  approximately: 
29a  5  bu.  to.l. 

299.  10  g  to  gr. 

300.  30  gr.  to  g. 

301.  10  bu.  to  1. 

302.  12  1  to  gal. 

303.  22  lb.  to  Kg. 

Reduce  approximately  : 

310.  16  Km  to  mi. 

311.  IG  Dl  to  bu. 

312.  75  1  to  qt 
3ia  32  1  to  bu. 

314.  40  1  to  gal. 

315.  10  mi.  to  Km. 

Reduce  approximately : 

322.  9  Kg  to  lb. 

323.  8  acres  to  a. 

324.  60  m  to  ft 

325.  4  m  to  in. 

326.  3  m  to  yd. 


304.  60  gal  to  1. 

305.  H4  1  to  gal. 

306.  64  1  to  bu. 

307.  15  gal.  to  1. 
30a  256  1  to  bu. 
309.  9  cords  to  8. 

316.  42  1  to  gal. 

317.  60  qt.  to  1. 
3ia  25  1  to  cu.  in. 

319.  10  1  to  cu.  in. 

320.  400  a  to  acres. 

321.  72  s  to  cords. 


327.  1  T.  of  water  to  cu.  ft. 
32a  6  tons  of  water  to  cu  m. 

329.  10  tons  of  water  to  cu  m. 

330.  10  cu  m  of  water  to  tons. 

331.  7500  sq.  miles  to  sq  Km. 


DENOMINATE    NUMBERS 


143 


EXACT  RESULTS  — WRITTEN  EXERCISES 

The  exact  equivalents  of  the  principal  units  should  be  memo- 
rized; the  exact  equivalents  of  the  other  units  should  be  com- 
puted from  the  principal  units. 


332.  Keduce  24  Km  to  mi. 
24  X  1000  X  S9.37  ^  ^,f^ 

12  X  S280 

24  Km  =14.9+  mi. 

333.  Reduce  5  Ha  to  sq.  dm. 
SxlOOx  100  X  100  =5,000,000 

6  Ha=^  5y000fi00  sq  dm. 


The  principal  unit  of  long  meas- 
ure is  the  meter,  39.37  in. ;  I  Km 
=  1000  m ;  1  mi.  =  6280  ft. 


The  principal  unit  of  land  meas- 
ure is  the  are,  100  sq  m ;  1  Ha  = 
100  a ;  1  sq  m  =  100  sq  dm. 


334.  Reduce  6  cu  Dm  to  cubic  yd. 

5  X  1000  X  39.37^  _  f,f.oQ  '7  I  '^^®  principal  unit  of  cubic  meas- 

— — — —         —00^i^./+       ^Jg  ig  I  Q^  m^    (39.37)»  cu.  in.; 

1728  cu.  in.  =  1  cu.  ft. ;  27  cu.  ft. 
5  cuJhn  =  6639.7+  cu.  yd.  =  1  cu.  yd. 


Reduce  365  cu  m  to  Dl. 
365  X  1000 


10 


=  36,500. 


365cum  =  36,500  Dl. 


The  principal  unit  of  capacity  is 
the  liter,  1  cu  dm.  1  cu  m  =  1000 
cudm. 


What  is  the  capacity,  in  cu.  ft.,  of  a  cistern  that  will  hold 
5  tonneaux  of  water?  Find  the  exact  answer  to  two  decimal 
places ;  the  approximate  answer. 

337.  6.4  Dl  of  potatoes  to  the  are  is  equivalent  to  how  many 
bushels  to  the  acre  ?  Find  the  exact  answer  to  two  decimal 
places ;  the  approximate  answer. 

33a  If  a  bushel  of  oats  weighs  32  lb.,  how  many  Kg  will  a 
HI  of  oats  weigh  ?  Find  the  exact  answer  to  two  decimal  places ; 
the  approximate  answer. 


144  DENOMINATE   NUMBERS 

MISCELLANEOUS 

339.  Change  20  mi.  70  rd.  6  ft  to  feet;  change  106,760  ft.  to 
integers  of  higher  denominations. 

340.  Express  }  rd.  in  yards  and  feet;  express  3  yd.  2  ft.  as  a 
fraction  of  a  rod. 

341.  Reduce  .3975  of  a  mi.  to  integers  of  lower  denominations ; 
reduce  127  rd.  3  ft  3.6  in.  to  the  decimal  of  a  mile. 

342.  Reduce  72  lb.  avoirdupois  to  lb.  troy;  reduce  87.5  lb. 
apothecaries'  to  lb.  avoirdupois. 

34a  Reduce  157.5  gal.  to  bu.  approximately;  reduce  16.8  bu. 
to  gal.  approximately. 

344.  Reduce  157.5  gal.  to  bu.,  exact  to  1  decimal  place ;  reduce 
16.9  bu.  to  gal.  exactly. 

345.  Reduce  38**  50'  30"  of  arc  to  time ;  reduce  2  hr.  35  min. 
22  sec.  of  time  to  arc. 

34e  Find  the  sum  of  18  gal.  3  qt,  and  6  gal.  3  qt  1  pt ;  find 
the  difference  between  25  gal.  2  qt  1  pt,  and  18  gal.  3  qt 

347.  To  Mar.  3,  add  182  da. ;  from  Sept.  1,  subtract  182.da. 

34a  To  Mar.  3,  add  6  mo.  2  da, ;  from  Sept.  5,  subtract  6  mo. 
2  da. 

349.  Find  the  time  from  the  discovery  of  America,  Oct  21, 
1492,  to  the  Declaration  of  Independence,  July  4,  1776. 

350.  Find  the  exact  number  of  days  from  the  fourth  of  July  to 
Christmas. 

351.  How  many  cu  cm  are  there  in  9  1  ?     1  in  9000  cu  cm  ? 

352.  Give  the  weight  in  Kg  of  5  cu  m  of  water ;  give  the  num- 
ber of  cu  m  in  5000  Kg  of  water. 

35a  How  much  does  a  grocer  gain  by  buying  3  bu.  of  chest- 
nuts at  $  3  a  bu.  dry  measure,  and  selling  for  5p  a  half  pint  liquid 
measure  ? 

354.  How  much  does  an  apothecary  gain  by  buying  50  lb.  of 
medicine  at  20^  a  lb.  avoirdupois  weight,  and  retailing  it  at  10^ 
an  ounce  apothecaries'  weight  ? 


LITERAL  QUANTITIES 


NOTATION  AND  NUMERATION 


A  number  may  be  considered  un- 
der conditions  that  are  directly 
opposed. 

One  of  the  conditions  is  regarded 
as  positive,  and  is  represented  by 
*-!-';  the  opposite  is  regarded  as  nega- 
tive, and  is  represented  by  *  —  /  The 
sign  *  +  '  has,  therefore,  an  arbitrary 
signification  ;  the  sign  *  —  '  denotes 
the  opposite  of  '  + '  in  the  same 
position. 

A  number  may  be  represented  by 
ft  letter,  and  this  letter  may  be  sub- 
jected to  the  various  operations. 

The  same  quantity,  base,  may  be 
used  several  times  as  an  addend. 
The  expression  is  abbreviated  by 
writing  the  base  and,  before  it,  a  num- 
ber, coefficient^  denoting  how  many 
times  the  base  is  used  as  an  addend. 
If  both  base  and  coefficient  are  num- 
bers, the  sign  '  x  *  is  necessary. 

AMKR.  ARITH.  —  10  146 


Illustrations 

6°  may  be  regarded  as  6*- 
above  zero,  or  6°  below  zero  ; 
6  mi.  may  be  regarded  as  6  mi. 
north,  or  6  mi.  south  ;  6  may  be 
regarded  as  6  to  be  added,  or  6 
to  be  subtracted. 

In  +  6°,  '  +  ♦  has  the  arbi- 
trary signification  above  zero  ; 
in  —  6°,  '  —  '  means  the  oppo- 
site, or  below  zero.  In  8  +  6, 
'  + '  has  the  arbitrary  significa- 
tion add;  in  8— 6,  '-'  means 
the  opposite,  or  subtract. 

Take  a  number,  z ;  multiply 
by  4,  4a:;  add  8,  4x  +  8;  di- 
vide by  2,  2  X  +  4  ;  subtract  2 
times  the  number,  4. 


a -f- a  +  a  +  a  =  4  a ; 

2  +  2-^2-1-2  =  4x2. 
In  the  former,  a  is  the  bast 
and  4  the  coefficient;   in  the 
latter,  2  is  the  base  and  4  the 
coefficient. 


146  NOTATTf>N    AM>   NUMERATION 

The  same  quantity,  busej  may  be 
used  several  times  as  a  factor.     The        «a<w=a*;  2x2x2x2=2*. 

expression  is  abbreviated  by  writing     '"  .^^^^  ^,^"°«''  «  »«  ^*\«  ^ 

'  ,  ,  .  "^    -  °      and  4   the  exponent:   in  the 

the  base  and,  over  it,  a  number,  ex-     ^^^^^^  2  Is  the  6a»e  and  4  the 

ponenty  denoting  how  many  times  the     exponent 

base  is  used  as  a  factor. 

1.  \Vhat  does  +  4  mean  ?     -  4  ? 

Since  *  +  *  has  an  arbitrary  signification,  we  may  assume  it  to  mean  any 
condition  which  has  an  opposite.  Thus :  +  4  may  mean  4  to  the  right,  4  up, 
4  to  the  north,  .... 

Since  ♦  -  '  means  the  opposite  of  *  +,♦  If  +  4  means  4  to  the  right,  -  4 
means  4  to  the  left ; .  . .  .  To  find  the  meaning  of  a  *  —  *  sign,  we  must  in- 
quire the  meaning  of  the  *  +  *  sign  in  the  same  position. 

2.  Write  a,  expressing  both  coefficient  and  exponent. 

An9.  +  1  a-^*.  When  either  coefficient  or  exponent  is  omitted,  '  +  P  is 
always  understood. 

a  If  *  +  6  mi.'  means  6  mi.  east,  what  does  '  —  8  mi.'  mean  ? 

4.  Analyze,  explain  the  meaning  of,  and  read,  Za?. 

3  o^  is  a  term  ;  3,  the  coefficient ;  a,  the  base ;  6,  the  exponent.  It  means 
a*  -f  o^  +  o^i  or  that  a  is  taken  5  times  as  a  factor,  and  that  the  result  is 
taken  3  times  as  an  addend.    It  is  read,  3,  a  to  the  fifth  power. 

5.  Write  in  simplest  form  that  2  is  used  6  times  as  an  addend ; 
that  a  is  used  6  times  as  an  addend. 

6.  Write  in  simplest  form  that  2  is  used  6  times  as  a  factor ; 
that  a  is  used  6  times  as  a  factor. 

7.  Write  in  simplest  form  that  a  is  used  6  times  as  a  factor, 
and  that  the  result  is  used  5  times  as  an  addend. 

a  Analyze,  explain  the  meaning  of,  and  read,  4  a?. 

9.  What  is  the  meaning  of  -|-  5  ?  of  —  5  ?  What  is  the  mean- 
ing of  *  -  $  6,'  if  ^  +  $6'  means  'icor^A  $6'? 


LITERAL   QUANTITIES  147 

10.  After  losing  3^,  a  boy  had  4^  left.     How  much  had  he  at 
first  ?     Analyze. 

11.  After  losing  a^,  a  boy  had  h^  left.     How  much  had  he  at 
first  ?     Analyze.  Ans.  (a  -\-  h)f. 

12.  After  losing  a  certain  sum,  a  boy  had  4^  left.     If  he  had 
7^  at  first,  how  much  did  he  lose  ?     Analyze. 

13.  After  losing  a  certain  sum,  a  boy  had  hf  left.     If  he  had 
a^  at  first,  how  much  did  he  lose  ?     Analyze.  Ans.  (a  —  h)f. 

14.  At  2^  each,  how  much  will  5  apples  cost  ?     x  apples  ? 

15.  At  6^  each,  how  much  will  x  apples  cost?       Ans.  hx  cents. 

16.  If  4  apples  cost  8^,  how  much  will  1  apple  cost?    If  x 
apples  cost  8^,  how  much  will  1  apple  cost? 

17.  If  X  apples  cost  6^,  how  much  will  1  apple  cost  ?     Ans.  -^. 

X 

la  At  4^  each,  how  many  apples  can  be   bought  for  8^? 
for  x^?    Analyze. 

19.  At  x^  each,  how  many  apples  can  be  bought  for  a^? 

20.  If  a  boy  had  x  marbles,  and  lost  y  of  them,  how  many  had 
he  left  ?  Ans.  (x  —  y)  marbles. 

21.  If  a  dog  runs  h  ft.  in  1  minute,  how  far  will  he  run  in 
C  min.  ?  in8min.  ?   in  a;  min.  ? 

22.  When  eggs  sell  at  x^  a  dozen,  what  is  the  selling  price  of 
each  egg?  Ans.  ^f, 

lib 

23.  In  Ex.  22,  what  is  the  selling  price  of  3  eggs  ? 

24.  If  eggs  cost  a^  each,  how  much  will  1  egg  cost  if  the  price 
is  increased  1^?  Ans.  (a  +  1)^. 

25.  If  3  eggs  sell  for  6^,  what  is  the  selling  price  of  4  eggs? 
of  5  eggs  ?   of  a;  eggs  ?  Ans.  2  x  cents. 

26.  How  many  horses  at  x  dollars  each,  must  a  man  sell  to  pay 
for  b  cows  at  c  dollars  each  ?  j       be  , 

x 

27.  The  product  of  the  sum  and  difference  of  4  and  2  is  4*  —  2*. 
Make  a  similar  statement  with  letters  instead  of  numbers. 


t 


148  ADDITION 

ADDITION 

I.  To  add  when  the  signs  are  alikey  write  the  sum  and  use  the 
common  sign;  to  add  when  the  signs  are  unlike,  write  the  difference 
and  use  the  sign  of  the  greater. 

Letustake  "^"^  ~^  +^  "^ 

4-2  -2  -2  4-2 

To  prove  the  sums       4-6  — 5  -fl  — 1 

In  these  examples,  let  us  assume  that  4  means  to  the  right,  and  -  to  the 

To  +  3  add  +2.     +3  means  3  to  the  right,  one,  two,  three 

p-| +  2  means  2  to  the  right,  one,  two;  counting,  we  have  6  to  the 

right,  or  +  6. 

To  —  3  add  +2.     —  3  means  3  to  the  left,  one,  two,  three ; 

i-f-  2  means  2  to  the  right,  one,  two ;  counting,  we  have  1  to  the 
left,  or  —  1. 
In  like  manner,  the  other  results  may  be  proved  to  be  —  5, 
and  +1. 

Examining  these  results,  we  see  that  there  are  two  cases ;  where 
the  signs  are  alike,  and  where  the  signs  are  unlike ;  and  that  the 
results  are  +5,  —  5,  4-1,  and  —  1. 

By  diagram,  find  tJie  sum  of: 

5  _3  _8  4-7  -4  -5 

2a   -6  -2  4-4  4-5  -3  4-3 

II.  To  add  quantities  having  a  common  factor,  add  the  factors 
not  common  and  retain  the  common  factor. 

Let  us  take  2  a 

To  prove  the  sum  8  a 

6  a  +  2  a  means  6  x  a  +  2  x  a,  or  that  a  is  taken  6  times  as  an  addend, 
and  then  2  times  more  as  an  addend ;  or  6  +  2,  or  8,  times  as  an  addend, 
or  8  a. 

Hence,  the  rule. 

NoTB.  —  The  pupil  should  compare  this  with  the  same  principle  on  p.  72. 


29. 

Add  -h  17  and 

+  17 

-20 

-   S 

31. 

Add  +6,-8, 

+  6 

-8 
-9 

+  7 

+  i5 
-17 

-  4 

LITERAL   QUANTITIES  149 

20.  30.   Add  -  12  and  -  8. 

-12 
-    8 


-20 

9,  -f7. 

The  sum  of  the  positive  addends  is  +  13 ; 
the  sum  of  the  negative  addends  is  —  17  ;  the 
sum  of  +  13  and  —  17  is  —  4. 

32.  Add  3a-26  +  c,  -a-h36-2c,  2a-36-4c. 

3a-26+    c 

—  a4-36  —  2c  The  sum  of  the  a's  is  4 a  ;  the  sum  of  the 

2  a  —  3  6  —  4  c  6's  is  —  2  6 ;  the  sum  of  the  c's  is  —  6  c. 


4a-26- 

-he 

NoTB.  —  Add  the  cohimns  from  the  left  in  their  order. 

AM: 

+  36 

-25 

-37 

+  82 

+  37 

3a     -47 

+  39 

-45 

-38 

+  39 

-64 

-66 

-54 

+  63 

-56 

34.      +38 

-32 

+  72 

-84 

-60 

AM: 

35.  -8,  -7,  +6,  +6,  -4,  -3,  +8,  +9,   -10,  +6. 

36.  5,  +6,  -7,  -5,  +8,   +9,  -7,  -8,  +12,  -3. 

AM: 

37.  7a+46  +  2c,  6a-36-2c,  -a  +  46-c,  a  -  6  +  c 
3a  3o«-26*-c»,  -a«  +  86*  +  2c»,  o«-46»-c>,  2a-36. 
39.  2a5«-2aj«  +  3a?,  -4«»  +  3aB*  +  a?,  Sir" -a^-3x,  -3iB*. 
4a  4a«6-3a6«  +  6c«,  2a«6  +  4a6«-36c«,  - 3 a«6  -  ae>«  +  6c». 


150  SUBTRACTION 

SUBTRACTION 

III.  To  suhtractj  change  the  si^n  of  the  subtrahend  and  proceed 
as  in  addition. 

Let  us  take  '^l  ~^  "^  +^ 

+  3  -3  4-3  -3 

To  prove  the  remainders  — 1  +1  — 5  +6 

In  these  examples,  let  us  assume  that  +  means  to  the  right,  and  —  to 
the  left. 

From  +  2  subtract  +  .3.     -f  2  means  2  to  the  right,  one,  two.    If  we  were 
to  add  +  3,  we  would  count  3  to  the  right,  but  since  sub- 

1 traction  is  the  opposite  of  addition,  we  must  count  3  to 

the  opposite  of  the  right,  or  3  to  the  left,  one,  two,  three ; 
counting,  we  have  1  to  the  left,  or  —  1. 

From  —  2  subtract  —  3.    —  2  means  2  to  the  left,  one, 
r^  1  two.    If  we  were  to  add  —3,  we  would  count  3  to  the  left, 

but  since  subtraction  is  the  opposite  of  addition,  we  must 
count  3  to  the  opposite  of  the  left,  or  3  to  the  right,  one,  two,  three  ;  count- 
ing, we  have  1  to  the  right,  or  +  1. 

In  like  manner,  the  other  results  may  be  proved  to  be  —  5  and  +  5. 

By  diagram^  subtract  the  lotcer  number  from  the  upper. 

2  6  3-4-8+3 

41.       3  -2  8  -6  -3  -5 

IV.  To  subtract  quantities  having  a  common  factory  subtract  the 
factors  not  common^  and  retain  the  common  factor. 

Let  us  take  ^^ 

6a 

To  prove  the  remainder  2  a 

8  a  —  6  a  means  8  x  a  —  G  x  a,  or  that  a  is  taken  8  times  as  an  addend, 
and  then  6  times  less  as  an  addend,  or  8  —  6,  or  2,  times  as  an  addend,  or 
2  a. 

Hence,  the  rule. 

Note.  —  The  pupil  should  compare  this  with  the  same  principle  on  p.  72. 


LITERAL  QUANTITIES  151 

42.   From  -  12  subtract  -  19.  43.  From  16  subtract  20. 

-12  16 

-19  _20 

T^  -4 

44.  From  a-2b-3c  +  5d  subtract  — a  +  Sb-^2c  —  2d. 

(I  —  -^  b  —  3c  -{-  od  ^  means  +  1  a  ;    —a  means  —la;  changing 

.a-\-'Sb-\-2c  —  2d      the  sign  of  the  subtrahend,  mentally,  and  adding, 


]a-5b-5c  +  7d     we  get  2  a,  etc. 
Subtract : 


17 

-72 

+  75 

-72 

+  81 

-86 

45.         18 

-80 

-85 

+  83 

-71 

+  73 

-50 

-35 

-56 

+  35 

-62 

-35 

46.    -80 

+  40 

-48 

+  60 

+  35 

+  62 

Subtract  : 

-4a 

-9a 

+  6a 

-8a 

+  7a 

-10a 

47.    H-5a 

-8a 
H-lGcZ^ 

—  7a 

+  6a 
-Urn' 

+  3a 
+  11  a^ 

-    7a 

-5c» 

+  18e 

-167l« 

4a    -f4c* 

-h    4cP 

-17e 

-16m» 

-    9a^ 

+  14n» 

Subtract  : 

49.  4a-66  +  10c  —  12d  from  26-2a  +  6c  +  2d. 

5a   2a:*  +  5a?y  +  a»  +  2y»  from  4«'  — 6a^  +  2a2!  +  2y". 

51.  4a;'-2ajV-f  3a^  +  23/»  from  3 «» +  3 a?»y - 4 ajy" - 3 3/». 

52.  3a^2/  +  9aV  +  6ary»-8  from  4ajV  +  8«y  -  6ajy»- 2. 

53.  -aV-3a*6c-2a6«c+6c»  from  aV+2a*6c+4a6«c+36c«. 


152  MULTIPLICATION 

MULTIPLICATION 

V.  The  product  of  like  signs  is  -f  ;  t?ie  product  of  unlike  signs 
is  —. 

Let  us  take  +3x-f2,  -3x-2,  +Sx-2,  -S  x  +  2. 

To  prove  the  products  -|-  6,  -h  6,  —  6,  —  6. 

+  3  X  +  2  means  that  +  3  is  taken  2  times  as  an  addend,  or  +  3  +  3, 
or  +6. 

—  3  X  —  2  means  that  —  3  is  taken  2  times  as  a  subtrahend,  or  —  (  —  3) 
-(-  3),  or  +  3  +  3,  or  +6. 

4-  3  X  -  2  means  that  +  3  is  taken  2  times  as  a  subtrahend,  or  —  ( +  3) 
-(+3),  or  -3-3,  or  -6. 

—  3  X  +  2  means  that  —  3  is  taken  2  times  as  an  addend,  or  ( -  3)  + 
(-3),  or  -3-3,  or  -6. 

Hence,  the  rule. 

Declare  the  products : 

54.  +6x  +  6;  -5x-6j  -6x  +  6;  4-6x-6. 

55.  -6x-7;   -6x  +  7;   -f6x-7;   -f-6x4-7. 

VI.  To  multiply  when  the  bases  are  the  same,  write  the  common 
hasey  and  over  it,  the  exponent  of  the  multiplicand  plus  the  exponent 
of  the  multiplier. 

Let  us  take  a*  x  a' 

To  prove  the  product        a* 


.8  — 


a  X  a 
a  X  a  X  a 


— /,« 


Analyzing  the  product,  we  see  that  the  base,  a,  is  the  common 
base ;  that  the  exponent,  5,  is  the  exponent  of  the  multiplicand 
plus  the  exponent  of  the  multiplier. 

Declare  the  products  : 

56.  a*xa^\  ai^xx']  f  xf;    b^xb-,   -  a^  x  -  a^;   -  a-^  x  +  a. 

57.  a^xa-j   a-xa^;  x*xa^',  y^  X  y,  +q^x-x-,  -{•  x^  x  -  x"- 


LITERAL  QUANTITIES  163 

SB,  Multiply   —8a  by  -3a*. 

The  product  of  like  signs  is  -f . 

—  o  *  To  multiply  when  the  bases  are  the  same, 

—  3  a*  write  the  common  base,  and  over  it,  the  expo- 
+  24" a*  ^^^^  ^^  ^^®  multiplicand  plus  the  exponent  of 

the  multiplier. 

59.  Multiply   a»  -  2  ab  +  b^  by  a-\-b. 

a*-2ab  +      6» 

a     4-       6  Beginning  at  the  left,  we  multiply  by  a; 

a»  —  2  a*6  +     ab*  a  x  a*  =  a';  a  x  -  2a6  =- 2a26  ;  ax&2=a62. 

„       ^    , ,       , ,  Then  we  multiply  by  6  ;  6  x  a*  =  a'^ft ;  6  X  — 

a»6-2a6'-f  &'  2ab  =  -2ab^;  bxb^  =  b». 
o?-    a*b-    ab^+b" 

Multiply  : 

eo.  a6  X  a6 ;  a'6  x  ab* ;   a^b  x  a*b^ ;  a^6'  x  a*6'. 

61.  -3ax-6a;    -36x26;   a*x-2a*. 

62.  -3a^zx2a:*;   2a5x3a^;    -SxyxSx'y^. 

63.  2a6x3a6;    -2a«6x3a6*;   2a26«x3a6. 

64.  —6xyzx2;    — 2x  —  6«y2;    — 2a^x3  xyhi. 


Multiply 
65. 

a+b  a-6  a-6  a*-2a6-h6' 


65.  e&  67.  6a 

a  +  6  a  —  h  a-\-b  a'4-2a6+6» 


69. 

3a  +  46 
3a-46 

70.* 

a^  +  a^ 
^-xy^ 

a*-l^ 
a«  +  6« 

72. 

^  +  xy  +  i^ 
T^-xy+y" 

73.  74.  75.  76. 

(a +  6)*  (o-f6)*  (a  4- 6)*  (a-\-b^cf 


164  DIVISION 

DIVISION 

VII.  The  quotient  of  like  signs  is  -k-  ;  the  quotient  of  unlike  signs 
is  — . 

-t-6  -6  -f6  -6 

Let  us  take  ^3'        ^3*       ZTg        4.3* 

To  prove  the  quotients     -f  2,        +2,        —  2,        —  2. 

+  8  x  +  2  =  +  6 
-8x+2=-6 
-8  x-2=  +  6 
+8x-2=-6 

Since  the  product  divided  by  either  factor  is  equal  to  the  other  factor, 

±-?--4.2-    Zl?-J.9.    ±_?~_2.     ~^-_9 

Hence,  the  rule. 

Declare  the  quotients : 

77.    +6-f-+2;     -6-!--2;     -|-6-t--2;     -6-f--h2. 

7a    -f8-^-2;     -8-J-  +  2;     -8-i--2;     +8-S-  +  2. 

VIII.  To  divide  when  the  bases  are  the  same,  write  the  common 
base,  and  over  it,  the  exponent  of  the  dividend  minus  the  exponent  of 
the  divisor. 

Let  us  take  c^-i-a^ 

To  prove  the  quotient  a* 

cfi  _a Xffxaxaxg 
a*  a  X  a  X  a 

=  a  X  a  =  a^ 

Analyzing  the  quotient,  we  see  that  the  base,  a,  is  the  common 
base ;  that  the  exponent,  2,  is  the  exponent  of  the  dividend  minus 
the  exponent  of  the  divisor. 

Declare  the  quotients  : 

79.  a*  ^  a* ;     a^  -i-a^;     a^-i-a-,     a*  -r-  a* ;     —  a^  -j-  —  a*. 

80.  a^^x;      x^-5-a^;     ic^-^ic*;    a^-^x*-,     +a^-^  — a;*. 


LITERAL  QUANTITIES 


166 


ai.   Divide  9  c*  by  —  Sc*. 

9c*-i 3c^  =  —  3  c*.  The  quotient  of  unlike  signs  is  — . 

To  divide  when  the  bases  are  the  same, 
write  the  common  base,  and  over  it,  the  exponent  of  the  dividend  minus  the 
exponent  of  the  divisor. 

82.  Divide  a*  +  b*-2  a'b''  by  h^-\-a?-2  ab. 

a*  -  2  aft  +  62)a*  -  2  a'^b'^  +  b\a^  +  2  a6  +  6« 
a*-2a»6  +  a^b^ 


+  2a'fe 
+  2a^b 


Ha^b-i+b* 
4  a^b^  +  2  ab* 


2  ab*  +  b* 
2  ab*  4-  6* 


Before  dividing,  it  is  necessary  to  arrange  the  terms  according  to  the 
descending  powers  of  some  letter. 

a*  is  contained  in  a*,  a* ;  a^  x  a"^  is  a* ;  a^  x  —  2ab  is  —  2  a'6 ;  a*  x  6'^ 
is  a^&'^ ;  subtracting  the  terms  in  the  order  of  the  descending  powers  of  a, 
+  2  a«6  -  3  a%^  +  b* ;  etc. 

NoTB.  — a<-fa'6  +  a2i2 -)-«58 -1-54  ig  arranged  according  to  the  descending 
powers  of  a.    The  expoueuts  decrease  in  order  from  the  largest. 

Divide : 


83. 

—  a'  -J-  —  a ; 

-6a«6-j-4-a6; 

—  4  oft'  -!-  aft. 

84. 

-a^  +  x»; 

_6ic«-j.-3a;«; 

-12x2y-!-2a^. 

85. 

_a^.^_a;5; 

-4a5--2a6; 

6  arV  H-  2  a;. 

86. 

ar*H--a^; 

6a^^H--2a6«; 

8ir*y2^-2xy. 

87. 

ar'-j--ar»; 

4a«6*^a»6»; 

6a^-hy'. 

Divide : 

88.  .r^-{-2xij-{-f  by  x-\-y]  a? -2xy  +  f  hy  x  —  y. 

89.  a^-y*        by  a;-y;  ar»  +  ^  by  a; -f- y. 

90.  a^  +  y*        by  x-f  y;  25a5«-20a^+4y*  by  6a;— 2y. 

91.  a^  +  aY  +  y*  by  7?-xy-\-%^\  aj»-y*  by  x-y. 

92.  6ar*-5a^-6y*  by  2aj-3y;         x^-y"  by  a:*-y». 


166  FACTORING 

FINDING  FACTORS 
Factor: 

9a  3a«6  +  3a6»;  6aj»- 12aj«  +  24»-6. 

94.  12oV-12a»6*;  14 aj^  -  28 a^  4- 7 icy  + 14. 

95.  24a>&»~36aV;  12«y-10ajy*  +  8a!y ~4ajV- 

96.  20a*6*-30a»;  10 aj^  -  20 aV  +  10 ay>  -  5 y*. 

97.  12a»6«  +  9a*6*;  18a?*y  +  30aV- 24aV-f- 12a^. 

Ex.  93.   8  ah{a  +  6).    8  a6  is  contained  in  8  a>6,  a  times ;  in  3  ah\  f>  timea 

GREATEST  COMMON   DIVISOR 
Find  the  O,  C,  D, : 

9a   Ixy,  Uf,  21y».  103.  15a*6,  20a«6»,  40(i^6*. 

99.   12«*,  6a^,  18  a^.  lOi.  6a^6*,  18a«6»,  2a<6. 

100.  9a^,  12  a^,  18  a?*.  105.  5a*6,  10a»6*,  a«6. 

101.  4y»,  16  y*,  22  y».  106.  9a*6«,  18a»6»,  21a«6». 

102.  12  a:*,  10  a^,  18.  107.   16a*6,  18  a^  12a»6». 
Ex.  9a.    Ana.  1  y.    Use  principle  III,  p.  76. 

LEAST   COMMON  MULTIPLE 
Find  the  L.  C.  M. : 

loa   2a%  Sa*b\  4ab\  113.  4ar»?/,  6a^,  3y*. 

109.   oa^b^y  6a%  Sab*.  114.  2xy,  oa^y,  3x. 

ua  Sa%  6a*6^  9a*6*.  115.  a;»,  2y«,  3y. 

111.  8a»6,  10a*6«,  12a*6^  ua  ar'^',  6/,  7  a;*. 

112.  ea%  5a»6*,  15a*6«.  117.  8a^,  12ar^y*,  16  a^. 
Ex.  108.    12  a*b*.    Use  principle  I,  p.  76. 


FRACTIONS— LITERAL  QUANTITIES  157 

REDUCTION 

16  a*b* 
lia  Reduce  ~ — -  to  lowest  terms. 
24  a6' 

16a*6*      2  a'  Dividing   both  terms  by  8a6*,  we 

24a6»^36'  obtain  |^.    See  p.  85. 

119.  Reduce  -^,  -— r,  to  fractions  having  the  same  de- 

36  9  0*^  15  ao 
nominator. 

30a«6     20  a       66  Th®  L.C.D.  is  i5ab^.     3  6  is  con- 

T? — T^'  T? — r»»  t:^ — r5  tained  in  the  L. C.  D.,  16 a6:  15 ab  x  2a 

46  a6*  45  ab^  4o  a6«  ^  g^  ^,^     ^^^  ^  ^^ 

ADDITION  AND   SUBTRACTION 

120.  Simplify  -  +  -• 

c     a 

a      c      a*  4-  c*  ^®  reduce  to   equivalent  fractions 

-  +  -  = •  having    the    L.  C.  D.,    and    unite    the 

^      "  ^  numerators.    See  pp.  88,  89. 

m.   Simplify  l^-2£^. 

2(4a;-5)-3(2a;-3)  j^  removing  -3(2  a: -3)  from  the 

6  parenthesis,  we  say,   —  3  times  2  x  = 

8«— 10— 6a:-|-9     2x—l  -Qx;    -3  times  -3  =  +  9,  because 

™              g              =      g      •  2  X  —  3  is  to  be  multiplied  by  —  3. 

MULTIPLICATION  AND  DIVISION 

122.  Simplify  — —^  x  — •  We  multiply  the  numerators  for  a 

Jo  6       3  a         new  numerator  and  the  denominators 
6o*       66 ___  2a  for  a  new  denominator,  canceling  when 

26  6«     3  a ""  6^*  possible.    See  p.  90. 

123.  Simplify     ^  ^^ 


8      X  ^^  ^'^*  =  —.  We  invert  the  divisor,  and  proceed 

9  «y         16  2  aa  iQ  multiplication.     /See  p.  52. 


158  FRACTIONS 

EXERCISES 
Reduce  to  lowest  terms: 

^^    6aVj^.    Sx^  75aW^.    96 ar».v» 

*  12a«6'  15a!V  '   126a^6V'  144  ary 

^^y  .     8ai»  32afec    .  72 xV 

*  24aY*  12aV  *  128a«W  2l6xy' 


Reduce  to  fractions  having  the  L,  C.  D, : 

-^    3x  2x  7x  5«  ,^   3x-2   6a;-4 

"^  T'  T'  y  T*  ^  -T-'  -Tr- 

ias. ?,  4  4?..  m.^--^^  «'-^ 


a  a^>'  a**  6«  8aj    '     12x 


^*nd  f^€  raiM«  of: 

132.  5^  +  ^£^.  134.  ^£:=I-2^z:i. 

6  9  3  2 

^^3^   5x  —  2y      7j:-3y  5x  — 3y     Sx—7y 

S       ^      12      '  8  12     * 


/^'nd  fAc  va7M€  o/; 

^25^     32/. 
-16  y"     35a^ 

137.  12^x^2_ 
55         15  y* 


32  aV 

44  a» 

^^    45  6^ 

■  356» 

139.    2-«\. 
125  if' 

236' 

LITERAL  QUANTITIES  169 

EQUATIONS 

IX.  To  transpose  a  term  from  one  member  of  an  equalion  to  the 
other,  change  its  sign. 

Illl'stratiox.     8  —  6  =  2;   transposing  —  6  to  the  right-hand  member, 
8  =  6  +  2. 

X.  Either  of  two  factors  is  equal  to  their  product  divided  by  the 
other. 

Illustration.    6  x  8  =  48 ;  whence  6  =  Y,  or  8  =  y . 

XI.  Multiplying  or  dividing  both  members  of  an  equation  by  the 

same  number,  cannot  affect  the  equality. 

Illustration.     12  =  8  +  4  ;  multiplying  both  members  by  2,  24=16+8  ; 
dividing  both  members  by  2,  6  =  4  +  2. 

XII.  Raising  both  members  of  an  equation  to  the  same  power,  or 

depressing  both  members  to  the  same  root,  cannot  affect  the  equality. 

Illustration,    jc*  =  4  ;  squaring  both  members,  x*  =  16  ;  extracting  the 
square  root  of  both  members,  x  =  2. 

Find  the  value  of  x: 

140.  6x-12  =  4x+-4.  147.  6aj*  =  46. 

141.  6  =  6x  +  15-8a;.  14a  6Va5  =  26. 

142.  9a;+-10-18  =  7aj.  149.  6a^  =  48. 

143.  7a;-10=:3a;+-6.  150.  6^  =  12. 

144.  33a;+-8-f-7=-61.  151.  3aJ*  =  48. 

145.  -17-8  +  30  =  -2aj.  152.  3\/x=6. 

146.  lox-4-+7x=20a?.  153.  2ar'  =  64. 

Ex.  140.  Transposing  to  the  left-hand  side  all  terms  which  contain  x, 
6x-4z  =  12  +  4;  uniting,  2  x  =  16  ;  x  =  8. 

Ex.  147.     6  X*  =  46  ;  x*  =  9  ;   extracting  square  root,  x  =  3. 
Ex.  150.     6v^  =  12 ;    v^  =  2  ;   cubing,  x  =  8. 

Note.  —  To  depress  a  term  to  the  cube  root,  is  t4»  fitui  one  of  the  three  equal 
factors  whose  product  is  that  term.  Thus,  the  cube  root  of  8,  written  v^,  is  2, 
because  2x2x2-8. 


160  PROBLEMS 

PROBLEMS 

The  solution  of  problems  by  the  literal  method  differs  from  the 
analysis  method  (see  p.  62)  in  that  the  relations  are  stated  directly, 
the  required  terms  are  represented  by  letters,  and  these  letters 
are  subjected  to  addition,  subtraction,  multiplication,  and  division. 

In  explaining  a  problem : 

After  every  statement  give  a  reason,  unless  the  reason  adds  nothing 
to  clearness. 

After  the  equation  is  ft>rmed,  do  not  explain  the  solution,  hut 
declare  the  result. 

In  the  proof f  state  the  first  relation  and  show  how  it  is  met;  state 
the  second  relation  and  show  how  it  is  met;  and  so  on. 

154.  The  sum  of  two  numbers  is  32,  and  one  of  them  is  3  times 

the  other.     What  are  the  numbers  ? 

Relations :  32  =  the  sum  of  two  numbers ;  the  larger  =  3  times  the 
smaller. 

First  Solution 

Let  X  =  the  smaUer,  8  Let  a;  =  the  smaller ;  then  3  x  must  equal  the 

3  a;  =  the  larger,  24  larger,  because  the  larger  is  3  times  the  smaller. 

4 a;  =  32  4x  =  32,  because  their  sum  is  32.     Whence 

jp  _  o  X  =  8,  the  smaller  ;  and  3  x  =  24,  the  larger. 

Second  Solution 
Let  X  =  the  larger,  24 

32  —  a;  =  the  smaller,  8  Let  x  =  the  larger ;  then  32  -  x  must  equal 

a;  =  3  (32  —  x)  *^®  smaller,  because  the  sum  of  the  two  is  32. 

__Qn       o  ^  —  3(32  —  x),  because  the  larger  is  3  times  the 

^  —  ^^  —  ^^  smaller.      Whence,    x  =  24,   the  larger ;    and 

4  X  =  96  32  -  X  =  8,  the  smaller, 

jc  =  24  Proof.  —  The  first  relation  is,  the  sum  of 

the  numbers  is  32  ;  24  +  8  =  32.     The  second 
PROOF  relation  is,  the  larger  is  3  times  the  smaller; 

1.24  +  8  =  32  24  =  3x8. 

2.  24  =  3  X  8 

Note.  —  There  is  no  rule  as  to  which  of  the  required  terms  shall  be  represented 
by  a:,  nor  as  to  which  of  the  relations  shall  be  used  first. 


LITERAL  QUANTITIES  161 

155.  There  are  54  children  in  a  schoolroom,  and  twice  as  many- 
boys  as  girls.     How  many  boys  are  there  ? 

15a  There  are  135  books  on  three  shelves ;  on  the  second  shelf 
there  are  twice  as  many  as  on  the  first,  and  on  the  third,  three 
times  as  many  as  on  the  second.  How  many  books  are  there  on 
each  shelf  ? 

157.  A  man's  property  is  worth  $3600.  His  barn  is  worth 
twice  as  much  as  his  house,  and  his  land  is  worth  as  much  as  his 
house  and  barn  together.     What  is  the  value  of  each  ? 

15a  A  farmer  has  208  animals,  consisting  of  horses,  sheep,  and 
cows.  What  is  the  number  of  each,  provided  that  the  number  of 
sheep  equals  three  times  the  number  of  cows,  and  the  number 
of  cows  equals  three  times  the  number  of  horses  ? 

159.  A  is  5  times  as  old  as  B ;  and  C,  3  times  as  old  as  B ;  the 
sum  of  their  ages  is  81  years.     What  is  the  age  of  each  ? 

16a  The  sum  of  $  264  was  raised  by  4  persons,  A,  B,  C,  and  D ; 
B  contributed  twice  as  much  as  A  ;  C,  4  times  as  much  as  A  and 
B  together ;  and  I),  one  half  as  much  as  B  and  C  together.  How 
much  did  each  contribute  ? 

161.  A  certain  fish  is  6  ft.  6  in.  in  length ;  its  tail  is  twice  as 
long  as  its  head,  and  its  body  is  as  long  as  its  tail  and  head 
together.    What  is  the  length  of  its  body  ? 

162.  Divide  96  into  3  parts,  such  that  the  first  part  shall  be 
three  times  the  second,  and  the  third  twice  the  sum  of  the  other 
two. 

163.  A  certain  number  is  expressed  by  three  digits  whose  sum 
is  12.  The  digit  in  hundreds'  place  is  twice  the  digit  in  units' 
place,  and  the  digit  in  tens'  place  is  three  times  the  sum  of  the 
other  two.     ^Vhat  is  the  number  ? 

164.  A  boy  has  30  pieces  of  money,  nickels,  dimes,  and  quar- 
ters; the  number  of  quarters  is  six  times  the  number  of  dimes; 
the  number  of  nickels  is  a  third  more  than  the  number  of  quar- 
ters.   How  much  money  has  he  ? 

▲MKH.   ARITH. —  11 


162 


PROBLEMS 


165.  Had  the  cost  of  a  horse 
been  three  times  as  much 
and  $70  more,  it  would  have 
been  9  445.  What  was  the 
cost? 


166.  If  63  is  subtracted  from 
a  number,  three  times  the  re- 
mainder will  he  twice  the  sum 
of  the  original  number  and  16. 
What  is  the  number  ? 


Relation :  3  thnes  cost  +  $70  - 

Relation :  3  x  (No.-63) =2  x  (No. 

$445. 

+  16). 

Let  a;  =  cost,  125 

Let  a;  =  the  No.,  221 

3  X  4-  70  =  supposed  cost 

X  — 63  =  the  rein. 

445  =  supposed  cost 

3  (a;  —  63)  =  three  times  rem. 

.-.  3a? +  70  =  445 

2  (a;  -f- 16)  =  twice  sum 

3a:  =  445-70 

.-.  3(a;-63)  =  2(a;4-16) 

3a;  =  375 

3a; -189  =  2a; +  32 

x=125 

a;  =  221 

Proof 

Proof 

1.  125  X  3  4-  70  =  446 

1.  3(221  -  63)  =  2(221  +  16) 

167.  One  number  is  three  times  another ;  if  I  take  the  smaller 
from  24  and  the  greater  from  46,  the  remainders  are  equal.  What 
are  the  numbers  ? 

16a  If  to  twice  a  certain  number  I  add  18, 1  obtain  120.  Find 
the  number. 

169.  Anna  is  four  years  younger  than  Mary;  if  three  times 
Anna's  age  is  taken  from  five  times  Mary's,  the  remainder  will  be 
62  years.     What  is  the  age  of  each  ? 

170.  Find  a  number  whose  excess  over  50  is  equal  to  twice 
what  it  lacks  of  being  113. 

171.  Divide  60  into  two  parts  such  that  one  part  may  exceed 
the  other  by  24. 

172.  The  joint  ages  of  a  father  and  son  are  70  years ;  if  the 
age  of  the  son  were  doubled,  the  result  would  be  five  years  more 
than  his  father's  age.     What  is  the  age  of  each  ? 


LITERAL  QUANTITIES  163 

173.  The  difference  between  two  numbers  is  28;  and  if  four 
times  the  less  is  added  to  the  greater,  the  sura  is  43.  What  are 
the  numbers  ? 

174.  The  sum  of  two  numbers  is  34 ;  the  larger  increased  by 
66,  is  9  times  the  other.     Find  the  numbers. 

175.  Divide  180  into  two  such  parts  that  one  of  them  dimin- 
ished by  35,  shall  be  equal  to  the  other  diminished  by  15. 

176.  After  30  gallons  had  been  drawn  out  of  one  of  two  equal 
casks  and  82  gallons  out  of  the  other,  there  remained  five  times 
as  much  in  one  cask  as  in  the  other.  What  was  the  capacity  of 
each  cask  if  both  were  full  at  first  ? 

177.  Divide  41  into  three  such  parts  that  the  second  shall  be 
4  more  than  the  first,  and  the  third  3  less  than  the  second. 

17a  A  man  borrowed  as  much  money  as  he  had,  and  then 
spent  ^  4 ;  he  then  borrowed  as  much  as  he  had  left,  and  spent 
$  3 ;  again  he  borrowed  as  much  as  he  had  left,  and  spent  $  2 ; 
he  then  had  nothing  left.     How  much  had  he  at  first  ? 

179.  An  estate  valued  at  $  4800  was  divided  in  such  a  manner 
that  the  wife's  share  plus  $400,  was  equal  to  three  times  the 
share  of  the  children.     What  was  the  wife's  share  ? 

180.  If  $300  is  subtracted  from  B's  income,  five  times  the 
remainder  will  be  three  times,  the  sum  of  $  3100  and  the  original 
income.     What  is  his  income  ? 

181.  A  merchant  began  business  with  a  certain  capital;  the 
first  year  he  doubled  it ;  the  second  year  he  gained  a  sum  equal 
to  the  original  capital  plus  $  100 ;  the  third  year  he  lost  as  much 
as  he  had  gained  the  first  year,  and  then  had  $  3100.  What  was 
his  original  capital  ? 

182.  A  lady  bought  two  pieces  of  cloth;  the  longer  lacked 
9  yards  of  being  three  times  the  length  of  the  shorter.  She 
paid  $2  per  yard  for  the  longer,  and  $3  for  the  shorter,  and 
the  shorter  piece  cost  as  much  as  the  longer.  How  many  yards 
were  there  in  each  piece? 


164  EQUATIONS  WITH   FRACTIONS 

EQUATIONS  WITH  FRACTIONS 

18a   Simplify  ^  +  ^  =  -^-1-6. 

18a;-|-6  =  —  10«-|-90  Multiplying  both  members  of 

1ft      I  in         QA      a  an  equation  by  the  same  number 

l»xH-lUa;-yu-b  ^^^  ^^^  ^^^^^  ^^^  equality. 

28  x  =  84  We  multiply  both  members  by 

^  =  3  theL.C.D.,  16.    ^xl6=18a;; 

?  X  16  =  6 ;    - ^-=  X  16  =  -  lOx;  6  X  16  =  90.    We  then  simplify  as  in  the 
6  3 

preceding  case. 

3-x     x-5      13 


184.   Simplify 


3  (3  —  a;)  —  2  (x  —  5)  =  39  we  multiply  both  members  by 

9-3x-2x+10  =  39  theL.C.D.,6.  3i:*x6=3(3-x), 

x  =  -4  =39. 


F«Md  x; 

185.^- 

3a:-4_-^ 
5 

190. 

5a;+l     3a;-2     lOar+1 
2             3              6 

5aj-4     1 
6          3 

191. 

3a;-2     4a;  +  5     25a; 
3*4           12 

"'•T- 

3a;4-2          1 
4              4 

192. 

a;  — 1      a;  +  l      a;  — 9 
5             6           10 

,««    4a; 

18a    ^- 

3x--f  2      33 
^4           4 

193. 

a;  +  l      a;-l      2a;-17 
6            5               5 

189.   i?- 
5 

3.T-2     29 
^4           4 

194. 

a;-2      a;-3      5a;-h3 
3             4             12 

LITERAL  QUANTITIES 


165 


PROBLEMS 


195.  Three  times  the  num- 
ber of  hours  before  nooi;  is  equal 
to  f  of  the  number  since  mid- 
night.    What  is  the  time  ? 

Relations :  3  times  hr.  before  noon 
=  \  hr.  since  midnight ;  hr.  since 
midnight  +  hr.  before  noon  =  12. 

Let  X  =  no.  hr.  since  mid.,  10 
12  —  a;  =  710.  hr.  before  noon,  2 

3(12-a:)  =  ^ 
5 

15(12 -«)=  3a; 

180-15x  =  3a; 

180  =  18a; 

a;  =  10 


196.  Divide  56  into  two  such 
parts  that  {  of  the  less,  di- 
minished by  \  of  the  greater, 
may  equal  12. 

Belations :  the  larger  number  + 
the  smaller  number  =  56;  f  of  the 
less  —  J  of  the  greater  =  12. 

Let  X  =  the  smaller,  24 

56  —  a;  =  the  larger,  32 

5  a;     56  —  x 

6  4 


=  12 


10a;- 3(56 -«)=  144 
10a;-168H-3a;=144 
13  a;  =  312 
a;  =  24 


Phoof 

1.  3x2  =  f  xlO 

2.  10 -f  2  =  12 


Proof 

1.  32  +  24  =  56 

2.  f  of  24- i  of  32  =  12. 

197.  In  a  school  of  495  pupils  there  are  J  as  many  boys  as  girls. 
How  many  girls  are  there  in  the  school  ? 

ISa  John  and  James  together  have  $  98 ;  if  James  has  J  as 
much  as  John,  how  mucli  has  each  ? 

199.  Mr.  Drake,  who  owns  f  of  a  tract  of  land,  has  14  acres  less 
than  Mr.  Brown,  who  owns  -j\  of  it.  How  many  acres  does  the 
whole  tract  contain  ? 

200.  In  a  certain  orchard  there. are  40  more  apple  trees  than 
peach  trees,  and  ^f  of  the  whole  number  are  peach  trees.  How 
many  peach  trees  are  there  ? 

201.  Jane's  age  is  }  of  Ann's,  and  the  sum  of  their  ages  equals 
21  years.     What  is  the  age  of  each  ? 

202.  Find  a  number  whose  f  part  exceeds  its  -j^  part  by  3. 


166  PR()BI>EMS 

203.  A  man  has  f  as  many  horses  as  cows,  and  the  cows  are 
15  more  in  number  than  the  horses.     How  many  horses  has  he  ? 

204.  The  width  of  a  room  is  j\  of  its  length ;  if  the  width  had 
been  4  feet  more  and  the  length  2  feet  less,  the  room  would  have 
been  square.     Find  its  dimensions. 

205.  I  bought  a  number  of  apples  at  the  rate  of  3  for  1  cent ; 
sold  one  third  of  them  at  2  for  a  cent,  and  the  remainder  at  5  for 
3  cents,  gaining  7  cents.     How  many  did  I  buy  ? 

206.  A  man  sold  a  horse  for  ^  of  its  cost,  plus  $  75,  and  thereby 
gained  $  15.     How  much  did  the  horse  cost  ? 

207.  Find  three  consecutive  numbers  such  that  if  they  are 
divided  by  7, 10,  and  17  respectively,  the  sum  of  the  quotients 
will  be  15. 

20&  A  man  bought  a  horse  and  carriage  for  $  280 ;  if  ^  the  price 
of  the  horse  is  subtracted  from  -J  the  price  of  the  carriage,  the 
remainder  will  be  the  same  as  if  1.9  times  the  price  of  the  carriage 
is  subtracted  from  2  times  the  price  of  the  horse.  What  was  the 
price  of  each  ? 

209.  Divide  $  115  between  two  men  so  that  f  of  what  the  first 
receives  shall  be  equal  to  J  of  what  the  second  receives. 

210.  B's  expenses  are  f  of  A's,  plus  f  30;  and  A's  expenses 
plus  ^  of  B's,  amount  to  $  795.     How  much  does  each  spend  ? 

211.  A  father  divided  $  1.43  among  his  three  sons  so  that  the 
first  had  f  as  much  as  the  second ;  and  the  third,  f  as  much  as 
the  first  and  the  second  together.  How  much  did  each  son  re- 
ceive ? 

212.  A  man  paid  $806  for  four  horses;  for  the  second  he  gave 
^  more  than  for  the  first;  for  the  third,  \  more  than  for  the 
second;  and  for  the  fourth  as  much  as  for  the  first  and  the 
third  together.     What  was  the  cost  of  the  fourth  horse? 

213.  A  farm  of  263  acres  was  divided  among  four  heirs,  so  that 
A  had  H  as  much  as  B ;  C,  as  much  as  A  and  B ;  and  D,  \  as 
much  as  A  and  C.     What  was  the  share  of  each  ? 


LITERAL  QUANTITIES 


167 


TWO  UNKNOWN   QUANTITIES 

If  we  have  two  equations  with  two  unknown  quantities,  the 
first  step  is  to  get  one  equation  with  one  unknown  quantity.  This 
step  may  be  taken  by  addition  and  subtraction,  by  substitution,  or 
by  comparison. 

Bt  Addition  and  Subtraction 


214. 


[2x  +  3y=^ 


4y=10 
18 


^  (^ ,  find  X  and  y. 
(2) 


6a;-8y  =  20 
6a;-i-9y  =  54 

17y  =  34 

y  =  2 


(3) 
(5) 


3aj-8  =  10 

3a;=18 

a;  =  6 

PROOF 

1.  18  -  8  =  10 

2.  12  +  6  =  18 

3x-43/  =  10 
2ajH-3y=18 


(1) 
(2) 


3a;  =  10  +  4y 


?*i5±M+3y=18   (3) 

o 

2(10 +  4y)  4-9.7  =  54 
20  4-8?/ +  9^  =  54 
17y  =  34 

y  =  2 

x  =  6 


We  may  eliminate  x  by  multiplying  (1)  by 
2,  and  (2)  by  3,  and  subtracting.  Multiply- 
ing (1)  by  2,  we  obtain  (3) ;  multiplying  (2) 
by  3,  we  obtain  (4). 

Subtracting  (3)  from  (4),  we  obtain  (6); 
whence,  y  =  2. 

Substituting  this  value  of  y  in  (1),  we  ob- 
tain 3  X  —  8  =  10 ;  whence,  x  =  6. 


By  Substitution 


We  may  eliminate  x  by  finding  the  value  of 
X  in  terms  of  y  in  (1),  and  substituting  tliis 
value  in  (2).     From  (1),  3x  =  10  +  4y,  and 

x  =  — i — ^;   substituting  this  value  of  x  in 

3 
(2),  we  obtain  (3). 

Clearing  (3)  of  fi-actions,  and  proceeding  as 
usual,  we  obtain  y  =  2. 

Substituting  and  proceeding  as  before,  we 
obtain  x  =  G. 


168 


TWO   UNKNOWN  QUANTITIES 


Bt  Compa&isom 


3x-4y=10    (1) 
2x  +  3y  =  18    (2) 

3 

18 


«  = 


Sy 


(3) 


10-h4y^l8-3y 
3  2 

2(10  H-4y)  =  3(18 -3y) 
20H-8y  =  54-9y 
17y  =  34 
y  =  2 
x  =  6 


We  may  eliminate  x  by  finding  the  value 
of  X  in  terms  of  y  in  (1)  and  (2),  and  plac- 
ing these  values  equal  to  each  other.     From 

(,),.  =  10±iJ!;from(2).x=li^; 

placing  these  values  equal  to  each  other,  we 
obtain  (3). 

Clearing  of  fractions  and  proceeding  as 
osuali  we  find  that  y  =  2,  and  x  =  6. 


Find  the  values  of  x  and  y : 

2a?-h3y  =  8, 
aJ  +  y  =  3. 


215. 


216. 


217. 


2ia 


219. 


220. 


r3a?-y  =  3, 
I4aj  +  y  =  ll. 

2x  +  3y  =  10, 

a;  +  2y  =  7. 

ic-y  =  4, 

2a;  +  y  =  14. 
|15x  +  8y  =  l, 
ll0a;-7y  =  -24. 

2x     3y_. 

■3""T-^' 

2     4 


221. 


222. 


223. 


224. 


225. 


226. 


r3a;  +  y  =  13, 
|ar-3y  =  -9. 


\x  +  y=^7, 

U- 

y  =  l. 

3ic 
2x 

+  2y  =  8, 

-y  =  3. 

[5x 
Sx 

-2/  =  4, 

+  y  =  4. 

Ix 
2y 

-3y  =  21, 
-5a;  =  -16. 

(5x 
3 

+  'f  =  94, 

Ix 
[  2 

5y_7 

3 

Note.  —In  examples  220  and  226,  the  first  step  is  to  clear  of  fractions. 


LITERAL  QUANTITIES  169 

227.  A  certain  number  expressed  by  two  digits,  is  equal  to  4 
times  the  sum  of  those  digits ;  if  27  is  added  to  the  number,  the 
digits  will  be  reversed.     Find  the  number. 

Relations  :  units'  digit  +  10  times  tens'  dij?it  =  number ;  4  (units'  digit  + 
tens'  digit)  =  number  ;  number  +  27  =  tens'  digit  +  10  times  units'  digit. 

Let  x  =  units*  digits  6 
y  =  tenf^  digit,  3 

a?  4- 10 1/  =  number  Let  x  =  units'  digit  and  let  y  =  tens' 

4  a;  4- 4  y  =  nwwifter  <^*6*^;  ^^^°  x  +  10y  =  the  number,  be- 

_i_in     —A       I  A  n\  cause  a  number  is  equal  to  its  units' 

.-.  X  +  luy  —  4a;  +  4y      (i;  ^j^j^.  ^  j^  ^.^^^^  ^^  ^j^^,  ^-^^^^  +  .  .  .  . 

a;  +  10?/  +  27  =  y4-10a;  (2)          4x  +  4y  =  the  number,  because  the 

X  —  Q  number  is  equal  to  4  times  the  sum  of 

_  „  its  digits  ;  x-\-\0y  =  4z  +  4y,  because 

y  —  *^  things  equal  to  the  same  thing  are  equal 

Proof  to  each  other ;  etc 

1.  4(3 +  6)=  36 

2.  36  +  27  =  63 

22a  The  sum  of  two  numbers  is  5  times  their  difference ;  twice 
the  greater,  increased  by  4  times  the  less,  is  56.    Find  the  numbers. 

229.  A  farmer  received  $  8.75  for  6  bu.  of  potatoes  and  5  bu.  of 
apples.  What  was  the  price  per  bushel  of  each,  if,  at  the  same 
rate,  7  bu.  of  potatoes  and  3  bu.  of  apples  were  sold  for  $  8.65  ? 

230.  A  man,  having  $  2.50  to  divide  among  a  certain  number  of 
boys  and  girls,  found  that  if  he  gave  each  of  them  10^,  he  would 
be  30^  out  of  pocket ;  so  he  gave  each  of  the  boys  8^  and  each  of 
the  girls  9^,  and  had  10^  left.     How  many  were  there  of  each  ? 

231.  A  certain  fraction  becomes  ^Ij  when  3  is  added  to  each  of 
its  terms,  but  becomes  J  when  3  is  subtracted  from  each  of  its 
terms.     Find  the  fraction. 

232.  If  A  gives  B  $  10,  B  has  twice  as  much  as  A  has  left ;  if 
B  gives  A  $30,  A  will  have  twice  as  much  as  B  has  left  How 
much  has  each  ? 


PROPORTION 


SIMPLE  PROPORTION  -  TERMS 


Division  may  be  expressed  by  writing 
the  dividend  before  and  the  divisor  ajXer 
a  colon.  Such  an  expression  is  a  ratio. 
See  p.  JfS, 

The  dividend  is  the  antecedent;  the 
divisor,  the  consequent. 

The  colon  is  read  *  is  to.* 

Two  fractions  may  be  equal;  in  the 
same  way,  two  ratios  may  be  equal,  a 
proportion. 

The  sign  of  equality  is  often  abbre- 
viated by  writing  the  extremities  of  the 
sign  *  =,'  thus  making  * : : '  read  *  as.' 

The  first  and  last  terms  of  a  propor- 
tion are  extremes;  the  second  and  third, 
means. 

The  means  of  a  proportion  may  be 
equal ;  then  each  is  a  7nean  proportional 
between  the  extremes. 


iLLUSTRATIOltV 

3  : 4,  ratio. 
Meaning,  3-4-4. 

3,  antecedent. 

4,  consequent. 
Z  is  to  4. 

I  =  f. 
3:4  =  6:8, 
a  proportion. 

3  :  4  : :  6  :  8, 
3  is  to  i  as  6  is  to  8. 

3  and  8  are  extremes  f 
4  and  6,  means. 

4:6  =  6:9. 
6,  a  mean  proportional 
between  4  and  9. 


Note.  — All  problems  in  proportion  may  be  solved  by  analysis.  The  pupil  is 
advised  to  master  simple  proportion,  but  to  use  analysis  for  the  solution  of  prob- 
lems which  fall  under  the  other  divisions  of  proportion. 

170 


PROPORTION  171 

I.  In  a  proportion,  the  product  of  the  extremes  is  equal  to  the 
product  of  the  means. 

«  «    J  «   ...        i^t  antecedent     Sd  antecedent       .      . 

Proof.     By  definition,  — =  -  r— ;  clearing  of  frac- 

J8t  consequent     M  consequent' 

tions,  1st  antecedent  x  Sd  consequent  =  1st  consequent  x  Sd  antecedent. 
Illustration.  3  :  4  :  :  9  :  12,   3  x  12  =  4  x  9. 

II.  If  three  teiinis  of  a  proportion  are  giveuy  the  other  term  may 
he  found. 

Proof.  Since  the  product  of  the  extremes  =  the  product  of  the  means, 
either  extreme  =  the  product  of  the  means  dividetl  by  the  other  extreme ; 
either  mean  =  the  product  of  the  extremes  divided  by  the  other  mean. 

Illustration.     If3:4=9:x,  x=  i^ ;   if  3  : 4  =  x  :  12,  x  =  ^^Ll?. 

3     '  '4 


III.  Tlie  mean  proportional  between  two  quantities  is  the  square 
root  of  their  product. 

Proof.  To  find  the  mean  proportional  between  two  numbers,  as  4  and 
9,  we  may  form  a  proportion  whose  extremes  are  4  and  9,  and  whose  means 
are  x  and  x.     Thus,  4  :  x  =  x  :  9 ;  whence  x"^  =  4  x  9,  and  x  =  Vi  x  9. 

1.  Write  a  fraction  whose  value  is  ^ ;  a  ratio  whose  value  is  J. 

2.  Write  a  proportion  each  of  whose  ratios  is  equal  to  | ;  each 
of  whose  ratios  is  equal  to  j. 

3.  Write  an  equation  which  is  an  equality  of  two  fractions; 
write  the  same  equation  as  a  proportion. 

4.  Is  4  :  8  : :  5  :  6  a  proportion  ?  Why  not  ?  W^hat  is  the  test 
of  a  proportion  ? 

5.  Find  the  mean  proportional  between  9  and  16;  3  and  12; 
4  and  25. 

S.  Define  ratio;  antecedent;  consequent;  proportion;  means; 
extremes. 


172  SIMPLE  PROPORTION 

PROBLEMS 

The  method  of  solving  a  problem  by  proportion  differs  from 
the  analysis  method,  in  that  each  relation  is  expressed  as  a 
proportion  whose  third  term  has  the  same  denomination  as  the 
answer,  and  whose  fourth  term  is  the  required  term.     See  p.  62. 

7.  If  2  apples  cost  8^,  how  much  will  3  apples  cost  ? 
Relation :  2  apples :  3  apples  =  cost  2  apples :  cost  3  apples. 

£:3  =  8:z  Since  the  answer  is  to  be  cents,  we  make  Sf 

g  X  S  ^®  third  term,  and  write  the  ratio  8f  :x. 

*  ~     ]j~  ~  ^^  Will  3   apples  cost  more  or  less  than  2 

Cost  3  aooleM  —  if  #  apples  ?      More  ;   then  we  make  the  second 

~        *  term  of  the  other  ratio  greater  than  the  first, 

and  we  have  5  apples: 3  apples  =  8f:x;  whence  x  =  12^,  the  cost  of  3 

apples. 

a  If  3  apples  cost  12^,  how  many  apples  can  be  bought  for  8^? 
Relation :  12  ^ :  8^  =  apples  for  12  f  :  apples  for  8^. 

li :  8  =  S :  X  Since  the  answer  is  to  be  apples,  we  make 

8  X  S  ^  apples  the  third  term,  and  write  the  ratio 

*  -    jg    =  *  8  apples :  x. 

No  atmles  =  f  ^^^^  ^  ^  ^"^  ™^'®  ®'  ^^^  apples  than  12  ^  ? 

Less ;  then  we  make  the  second  term  of  the 
other  ratio  less  than  the  first,  and  we  have  Iff:  8f  =  S  apples  :x;  whence 
X  =  2,  the  number  of  apples. 

9.  If  3  men  can  do  a  piece  of  work  in  12  days,  how  many  days 
will  2  men  require  ? 

Relation :  2  men  :  3  men  =  days  for  3  men  :  days  for  2  men. 

g:  S  =  IS :x  Since  the  answer  is  to  be  days^  we  make 

S  X  12  ^2  days  the  third  term,  and  write  the  ratio 

*  =  -^— =^^  ISdays-.x. 

No  davs  =  18  ^'^^  ^  ^^'^  require  more  or  less  days  than 

3  men  ?  More ;  then  we  make  the  second  term 
of  the  other  ratio  greater  than  the  first,  and  we  have  2  men :  S  men 
—  IS  days :  x ;  whence  x  =  18,  the  number  of  days. 

Note.  —  Ask  about  the  term  in  the  question,  in  the  denomination  of  the 
answer. 


PROPORTION  173 

10.  How  much  will  54  bu.  of  potatoes  cost  if  16  bu.  cost  $  5.60  ? 

11.  How  many  eggs  can  be  bought  for  70^  at  the  rate  of  2 
lor  o^? 

12.  If  16  men  can  dig  a  trench  in  24  days,  how  many  men  will 
be  required  to  dig  it  in  32  days  ? 

13.  If  16  men  can  dig  a  trench  in  24  days,  how  many  days  will 
12  men  require  ? 

14.  If  $150  gains  $12  in  8  months,  in  what  time  will  it  gain 
$17? 

15.  If  a  man  travels  32  miles  in  4  hours,  how  many  miles  can 
he  travel  in  5  hours  at  the  same  rate  ? 

16.  A  flagstaff  82  ft.  high  casts  a  shadow  62  ft.  long.  Under 
the  same  conditions,  what  must  be  the  height  of  a  steeple  which 
casts  a  shadow  93  ft.  long  ? 

17.  If  it  costs  $12  to  carpet  a  room  with  carpet  4  ft.  wide, 
how  much  will  it  cost  if  the  carpet  is  3  ft.  wide,  provided  there 
is  no  waste  and  the  cost  per  linear  yard  is  the  same  ?    See  Ex.  31. 

la  How  many  bushels  of  wheat  can  be  bought  for  $102.12, 
if  24  bu.  can  be  bought  for  $  13.32  ? 

19.  If  a  man  gains  $  1500  from  his  business  in  1  yr.  6  mo., 
how  much  will  he  gain  in  3  yr.  9  mo.  at  the  same  rate  ? 

20.  A  garrison  of  600  men  has  provisions  for  80  days;  how 
many  men  must  leave  to  make  the  provisions  hold  out  20  days 
longer  ? 

21.  A's  rate  of  working  is  to  B's  as  3:5.  How  long  will  it 
take  B  to  do  what  A  does  in  48  days  ? 

22.  If  40  men  can  build  a  wall  in  6  days,  at  the  same  rate 
how  long  will  it  take  16  men  to  do  |  the  same  work  ? 

2a  If  3  men  can  do  a  piece  of  work  in  51  days,  how  many 
men  must  be  added  to  the  number  to  do  the  work  in  17  days  ? 

24.  James  can  do  2\  times  as  much  work  in  a  given  time  as 
John.  How  long  will  it  take  James  to  do  what  John  does  in 
38  hours? 


174  COMPOUND   PROPORTION 

COMPOUND  PROPORTION 

A  compound  fraction  is  a  fraction  of  a  fraction ;  in  the  same 
way,  a  comjjound  ratio  is  a  ratio  of  a  ratio. 

To  find  the  value  of  a  compound  fraction,  we  multiply  the 
numerators  for  a  new  numerator  and  the  denominators  for  a  new 
denominator ;  to  find  the  value  of  a  compound  ratio,  we  multiply 
the  antecedents  for  a  new  antecedent  and  the  consequents  for  a 
new  consequent 

A  compound  proportion  is  a  proportion  having  one  or  both  of 
its  ratios  compound. 


25.   Find  the  value  of  the  compound  ratio,  *"     [ »  o^  «  *     [  >  o^ 
^S0:2l)- 


0:12 
3:    2 


4:7 
26.  In  the  compound  proportion,  9 : 6 

14:18 


=2 :  X,  find  the  value  of  x. 


J  7  xgx  ^^x^— •  The  product  of  the  extremes  is  equal  to 

""    4x  9  \  14     ""  the  product  of  the  means. 

27.   If  2  men  in  14  da.  of  10  hr.  each  earn  $  280,  how  many  hr. 

per  da.  must  3  men  work  to  earn  $  120  in  5  da.  ? 

S:  g     ]  Since  the  answer  is  to  be  hr.per 

280: 120  \  =  10:z  da.y  we  make  10  hr.  the  third  term, 

5: 14   -'  and  write  the  ratio  10  hr. :  x. 

— S  X  280  X  5 -^'^^'P^^-       hr.  per  da.  than   2   men?    Less; 

then  we  make  the  second  term  of 
the  other  ratio  less  than  the  first,  and  we  have  S  men :  2  men. 

Will  $120  require  more  or  less  hr.  per  da.  than  $280  ?  Less ;  then  we 
make  the  second  term  of  the  other  ratio  less  than  the  first,  and  we  have 
$280:^120. 

Will  5  days  require  more  or  less  hr.  per  da.  than  14  days  ?  More  ;  then 
we  make  the  second  term  of  the  other  ratio  greater  than  the  first,  and  we 
have  5  days :  14  days. 

NoTK  1.  — We  compare  each  set  of  ratios  with  the  second  ratio  separately. 
We  ask  about  the  term  in  the  question,  in  the  denomination  of  the  answer. 

Note  2.  —  Elxamples  in  Compound  Proportion  are  found  on  p.  177. 


PROPORTION  175 

ANALYSIS   METHOD 

It  is  recommended  that  Analysis  be  substituted  for  Simple 
Proportion. 

2a  If  2  apples  cost  8^,  how  much  will  3  apples  cost?  See 
Ex.  7. 

Relation :  3  apples  will  cost  f  as  much  as  2  apples. 

-x8  =  IX  Since  3  apples  will  cost  §  as  much  as  2 

apples,  3  apples  will  cost  |  of  8^,  or  12^. 
Coat  3  apples  =  Igf. 

29.  If  3  apples  cost  12^,  how  many  apples  can  be  bought  for 
«^?     See  Ex.  8. 

Relation :  Sf  will  buy  ^^  as  many  apples  as  12^. 

Jg^  Since  8  f  wiU  buy  ^  as  many  apples  as  12  ^, 

Sf  will  buy  ^j  of  3  apples,  or  2  apples. 
No.  apples  =  f . 

30.  If  3  men  can  do  a  piece  of  work  in  12  days,  how  many  days 
will  2  men  require  ?     See  Ex.  9. 

Relation :  2  men  will  require  |  as  many  days  as  3  men. 

Ix  IS  =  18  Since  2  men  will  require  |  as  many  days  as 

£  3  men,  2  men  will  require  f  of  12  days,  or  18 

No.  days  =18.  ^^y^' 

31.  If  it  costs  $  12  to  carpet  a  room  with  carpet  4  ft.  wide,  how 
much  will  it  cost  if  the  carpet  is  3  ft.  wide,  provided  there  is  no 
waste  and  the  cost  per  linear  yard  is  the  same  ?     See  Ex.  17.    . 

Necessary  knotcledge :  carpete  are  sold  by  the  linear  yard ;  the  less  the 
width  of  the  carpet,  the  greater  the  length. 

Relation :  for  the  room,  carpet  3  ft.  wide  will  cost  |  as  much  as  carpet 
4  ft.  wide. 

i.x  lt=  16  Since,  for  the  room,  carpet  3  ft.  wide  will 

S  cost  \  as  much  as  carpet  4  ft.  wide,  the  3  ft 

Cost  =  $  16.  carpet  will  cost  J  of  9 12,  or  $  10. 

NoTK.  —  The  pupil  should  solve  the  examples  on  p.  173  by  this  method. 


176  ANALYSIS  METHOD 

It  is  recommended  that  Analysis  be  substituted  for  Compound 
Proportion. 

32.  If  2  men  in  14  da.  of  10  hr.  each  earn  $  280,  how  many 
hr.  per  day  must  3  men  work  to  earn  $  120  in  5  days  ? 

^x  —  x^xiO  =  «  ^^^^^^^-   3  ™^°  ^^^^  require  |  as  many 

5     280      5  hr.  per  day  as  2  men  ;  to  earn  i  120  will  require 

Ans   Shr.perda.  U8a8ma»y '^'••/>«rd«ya8toeam  ^280;  5days 

*  will  require  V  as  many  hr.  per  day  as  14  days. 

Since  the  first  set  require  10  hr.  per  day,  the  second  will  require  |  x  ^f^ 

X  V  ><  ^^  ^^-  P^*"  ^*^y>  ^^  ^  hr.  per  day. 

NoTB.  —  In  such  problems,  the  relation  is  more  plainly  seen,  if  we  ask  aboat 
the  term  in  the  question^  in  the  denomination  of  the  answer.  Thus:  will  3  men 
require  more  or  less  hr.  per  day  than  2  men  ?  Less,  i  as  many ;  will  6 120  require 
more  or  less  hr,  per  day  than  S28D?    Less,  US  as  many  .... 

3a  If  5  compositors,  in  16  days,  of  8  hours  each,  can  compose 
20  sheets  of  24  pages  in  each  sheet,  50  lines  in  a  page,  and  40 
letters  in  a  line,  in  how  many  days,  of  4  hours  each,  will  10  com- 
positors compose  a  volume  to  be  printed  in  the  same  type, 
containing  40  sheets,  16  pages  in  a  sheet,  60  lines  in  a  page,  and 
50  letters  in  a  line  ? 

f  V  A  V  -^  V  ^«  V  ^<^  V  ^^  V  r/:  -  t#  Belations:  4  hr.  per  day  will 

4     10     20     24     50     40  require  \  as  many  days  as  8  hr. 

.         ^^  _  per  day :  10  compositors  will  re- 

Ans.   Sz  days.  .       t  -,  c 

'  quire  /^  as  many  days  as  5  com- 

positors ;  40  sheets  will  require  f  g  as  many  days  as  20  sheets ;  16  pages  will 
require  J|  as  many  days  as  24  pages ;  60  lines  will  require  f  g  as  many  days 
as  60  lines ;  60  letters  will  require  |§  as  many  days  as  40  letters. 

Since  the  first  task  requires  16  days,  the  second  will  require  |  x  /j  x  |g 
X  if  ><  f  M  IB  X  16  days,  or  32  days. 

34.  If  9  men  can  perform  a  certain  labor  in  17  days,  how  long 
will  it  take  3  men  to  do  twice  as  much  ? 

35.  If  a  man  working  6  hours  a  day,  6  days  in  a  week,  and  42 
weeks  in  a  year,  earns  $  1323,  how  much  mil  he  earn  if  he  works 
8  hours  a  day,  5  days  in  a  week,  and  50  weeks  in  a  year  ? 


PROPORTIOX  177 

36.  If  25  men  can  build  a  wall  200  ft.  long  in  40  days,  how 
many  days  will  4  men  require  to  build  a  similar  wall  600  ft. 
long? 

37.  If  12  iron  bars  4  ft.  long,  3  in.  wide,  and  2J  in.  thick,  weigh 
1050  lb.,  what  is  the  weight  of  26  iron  bars  6  ft.  long,  4  in.  wide, 
and  2  in.  thick  ? 

3a  If  36  men  can  dig  a  trench  150  ft.  long,  6  ft.  wide,  and 
4  It  6  in.  deep,  in  24  days,  how  many  days  will  32  men  require 
for  a  trench  210  ft.  long,  5  ft.  wide,  and  4  ft.  deep  ? 

39.  If  5  men  in  6  days  of  8  hours  each  can  mow  16  acres, 
how  many  days  of  6  hours  each  must  10  men  work  to  mow  24 
acres  ? 

40.  What  is  the  weight  of  a  block  of  stone  10  ft.  6  in.  long, 
6  ft.  8  in.  wide,  5  ft.  3  in.  thick,  if  a  block  of  the  same  stone  8  ft. 
long,  3  ft.  6  in.  wide,  and  3  ft.  thick,  weighs  12,600  lb.  ? 

41.  If  20  men  build  240  rd.  of  fence  in  24  da.  of  9^  hr.,  how 
many  hours  a  day  will  16  men  have  to  work  to  build  300  rd.  of 
fence  in  50  da.  ? 

42.  If  a  bin  16  ft.  long,  6  ft.  wide,  and  4  ft.  high,  holds  308  bu. 
of  grain,  what  must  be  the  height  of  a  bin  24  ft.  long,  5  ft.  4  in. 
wide,  to  hold  462  bu.  ? 

43.  If  there  is  no  waste  in  either  case,  how  much  will  it  cost 
to  cari)et  a  room  with  carpet  3  ft.  wide  at  90^  per  yard,  if  it  costs 
$  25  to  carpet  the  same  room  with  carpet  4  ft.  wide  at  $  1.25  per 
yard  ? 

44.  In  the  reprint  of  a  book  consisting  of  810  pages,  50  lines, 
instead  of  40,  are  contained  in  a  page,  and  72  letters,  instead  of 
60,  in  a  line.     Of  how  many  pages  will  the  new  edition  consist? 

45.  If  15  men  cut  480  steres  of  wood  in  10  days  of  8  hours 
each,  how  many  boys  will  it  take  to  cut  1152  steres,  only  J  as 
hard,  in  16  days  of  6  hours  each,  provided  that,  while  working,  a 
boy  can  do  only  |  as  much  as  a  man,  and  that  \  of  the  boys  are 
idle  at  a  time,  throughout  the  work  ? 

AMKR.    ARITU.  —  12 


178  ANALYSIS   METHOD 

It  is  recommended  that  Analysis  be  substituted  for  Partitive 
and  for  Conjoined  Proportion. 

46.  Divide  360  into  three  parts  proportional  to  3,  4,  and  6. 

Ji  0/  360  =  90,  first  Relations :  first  given  part  =  ^,  of  the 

jt  sum  of  the  given  parts ;  second  given  part 

^  of  360  =  ISO,  secotid  =  ^  of  the  sum  of  the  given  parts  ;  third 

J.  given  part  =  A  of  the  sum  of  the  given 

^  of  360  =  250,  third  parts. 

Therefore,  first  required  part  =  ^^  o^  the 

^^^^^  gum  of  the  required  parts,  or  ^^  of  360,  or 

1.  ^  +  ISO  +  160  =  360  90  ;  second  required  part  =  i^  of  the  sum 

2.  90 :  ISO  =  3:4  of  the  required  parts,  or  ^  of  360,  or  120  ; 

3.  90 :  150  =  3:6  etc. 

4.  ISO:  150  =  4 '6 

47.  A,  B,  and  C  go  into  partnership.  A  puts  in  $960;  B, 
$510;  and  C,  $1440.  If  they  gain  $  727.50,  how  much  should 
each  receive? 

960 
^'«  =  ^^  X  7S7.60  =  S40  nelations :  A's  investment  =  ^%^^  of  the 

whole  investment ;  B's  =  /^^  of  the  whole ; 

B's  =  ^  X  7S7.60  =  127.60       C's  =  ^ fg  of  the  whole. 

^^^^  Therefore,  A's  gain  =  /^  of  the  whole 

C'«  =  M^  X  7S7.50  =  360  gain,  or  /^  of  «  727.60,  orij  240  ; 


48.  If  10  barrels  of  apples  are  worth  7  cords  of  wood,  and  14 
3ords  of  wood  are  worth  5  tons  of  hay,  how  many  barrels  of 
apples  are  worth  50  tons  of  hay  ? 

5(?  X  ^  X  —  =  SOO  Belations :  1  ton  of  hay  is  worth  i^  cords 

5       7  of  wood  ;  1  cord  of  wood  is  worth  J^  bbl.  of 

Ans.  SOO  bbl.  apples.  ^PPjf-   ^  u  i.      ,o  w, 

Therefore,  1  ton  is  worth  i^  x  V   hhl., 

and  50  tons  are  worth  60  x  ^5*  x  V-  bbl.,  or  200  bbl. 

49.  Divide  $510  into  three  parts  which  shall  be  to  each  other 
as  2,  3,  and  5. 


PROPORTION  170 

50.  An  insolvent  debtor  fails  for  .f  3780,  and  is  able  to  pay  only 
$1550.     If  A's  claim  is  $378,  how  much  will  he  receive  ? 

51.  Divide  $  873  among  A,  B,  and  C,  so  that  for  every  $  2  that 
A  receives,  B  shall  receive  $4,  and  C,  $3. 

52.  A  and  B  buy  goods  to  the  amount  of  $600,  of  which  A 
pays  $250,  and  B,  $350.  If  they  lose  $150,  what  will  be  the 
loss  of  each? 

5a  A  bankrupt  owes  A  $  600 ;  B,  $  800 ;  C,  $  1000 ;  I),  $  1200  ; 
but  his  property  is  worth  only  $  1440.  How  much  should  each 
of  his  creditors  receive  ? 

54.  Divide  a  man's  estate  of  $29,000  so  that  his  wife  shall 
receive  $7  for  every  $5  received  by  each  of  his  two  sons,  and 
every  $  4  received  by  each  of  his  three  daughters. 

55.  A,  B,  and  C  engage  in  business,  A  putting  in  $  982 ;  B, 
$365;  and  C,  $843.  If  their  profit  in  one  year  is  $1460,  what 
is  each  one's  share  ? 

56.  A,  B,  and  C  plant  1200  acres  of  corn,  A  planting  2  times  as 
many  acres  as  B ;  and  B,  3  times  as  many  acres  as  C.  They  sell 
the  entire  crop,  amounting  to  45  bu.  to  the  acre,  at  22^  per  bu. 
What  is  each  man's  share  of  the  profit  ? 

57.  If  10  lb.  of  cheese  are  equal  in  value  to  7  lb.  of  butter,  and 
14  lb.  of  butter  to  5  bu.  of  corn,  and  12  bu.  of  corn  to  8  bu.  of  rye, 
how  many  pounds  of  cheese  are  equal  in  value  to  4  bu.  of  rye  ? 

5a  If  15  bu.  of  wheat  are  wortli  18  bu.  of  rye,  and  5  bu.  of  rye 
are  worth  8  bu.  of  corn,  and  9  bu.  of  corn  ai'e  worth  12  bu.  of 
oats,  and  16  bu.  of  oats  are  worth  20  lb.  of  coffee,  how  many 
pounds  of  coffee  should  be  exchanged  for  20  bu.  of  wheat  ? 


SOLUTION   OF  PROBLEMS 


METHODS  OF  PROCEDURE 


There  are  three  methods  of 
solving  problems :  analysiSj  the  lit- 
eral metfiodf  and  proportion.  In 
each,  the  relations  between  the 
given  terms  and  the  required 
terms  are  expressed  by  equations. 

By  analysis,  the  required  term 
must  form  one  member  of  an 
equation,  and  the  given  terms  the 
other;  no  operation  is  performed 
upon  the  required  term. 

By  the  literal  method,  no  attempt 
is  made  to  place  the  required  term 
by  itself,  but  the  equation  is  stated 
naturally,  and  the  operations  are 
performed  upon  the  terms  without 
discrimination. 

By  proportion,  the  equation  is 
stated  as  an  equality  of  two  ratios, 
the  antecedent  of  the  second  ratio 
being  of  the  same  denomination 
as  the  answer,  and  the  consequent, 
the  required  term. 

180 


Illcsthations 

At  4f  each,  how  many  apples 
can  be  purchased  for  8^  ? 


Analysis 

Belation :  number  of  apples  = 
the  number  of  times  cost  of  1 
apple  is  contained  times  in  cost 
of  all. 

Solution:  since  1  apple  costs 
4f,  as  many  apples  can  be  bought 
for  Bf,  as  4*  is  contained  times 
in  8f ,  or  2  apples. 

Literal  Method 
Belation :  cost  of  all  =  cost  of 

1  apple  X  no.  of  apples. 

Solution :  let  x=no.  of  apples ; 

4x  =  cost  of  all ;  4 x  =  8 ;  x  =  2, 

no.  of  apples. 

Proportion 
Belation :  cost  of  1 ;  cost  of  all 
=  1  apple  :  all  apples. 

Sxj 
4 


Solution  .-4:8 
2,  no.  apples. 


1  :  x:  x=- 


SOLUTION  OF  PROBLEMS 


181 


PROBLEMS  OF  PURSUIT 

Analysis  proceeds  indirectly,  introducing  a  special  method  for 
each  special  case;  the  literal  method  proceeds  directly,  expressing 
the  required  term  by  a  letter,  which  is  used  as  a  number.  Both 
methods  should  be  mastered ;  the  former  will  give  power  for  an 
indirect,  and  the  latter  for  a  direct,  attack  upon  a  problem. 

1.  At  what  time  between  3  and  4  o'clock  are  the  hands  of  a 
watch  opposite  to  each  other  ? 

Necessary  knowledge:  at  3  o'clock  the  min.  hand  is  at  12,  and  the  hr. 
hand  at  3;  when  the  hands  are  opposite  to  each  other,  they  are  30  min. 
spaces  apart ;  while  the  min.  hand  advances  60  spaces,  the  hr.  hand  advances 
5  spaces. 


Analysis 

Relation :  the  min.  hand  will  ad- 
vance as  many  min.  spaces  as  the 
number   of    spaces   it 
gains  in  1  min.  is  con- 
tained   times    in    the 
1^  number  of  spaces  to  be 
fC    gained. 

The  min.  hand  has 
advanced  from  ^  to  D,  or  46  spaces 
+  B  to  C;  because  from  ^  to  B  is  16 
spaces,  and  from  C  to  D  is  30  spaces. 
The  hr.  hand  has  advanced  from  B 
to  C;  therefore,  the  min.  hand  has 
gained  46  spaces. 

Since  the  min.  hand  advances  60 
spaces  while  the  hr.  hand  advances 

6  spaces,   the  min.   hand  gains  65  Proof 

spaces  in  60  min.,  or  y,  or  \\  of  a 

space  in  1  min.     Since  it  gains  \\ot  1.  \9^  =  12  x  4^*^. 

aspace  in  1  min.,  it  will  take  as  many  2.  40 j^  =  46 -H  4  j^. 

min.  to  gain  46  spaces,  as  }|  is  con- 
tained times  in  46,  or  49 ^^  min. 

2.  How  many  minute  spaces  does  the  minute  hand  of  a  watch 
gain  on  the  hour  hand  in  1  minute  ? 


Literal  Method 

Relations :  spaces  passed  by  min. 
hand  =  12  times  spaces  passed  by 
hr.  hand ;  spaces  passed  by  min. 
hand  =  spaces  passed  by  hr.  hand  + 
spaces  gained  by  rain.  hand. 

Let  X  =  sp.  pas.  hr.  A.,  4j^ 

12x  =  »p.  pas.  min.  A.,  49^ 

X  +  45  =  sp.  pas.  min.  h. 

.-.  12x  =  a;  +  45 

12x-x  =  46 

llx  =  45 

x  =  4^ 


182  PROBLEMS  OF  PURSUIT 

3.  If  the  three  hands  of  a  watch  all  turn  on  the  same  point, 
how  many  minute  spaces  does  the  second  hand  gain  on  the  hour 
hand  in  1  minute  ? 

4.  When  the  hands  of  a  watch  are  first  20  min.  spaces  apart 
between  5  and  6  o^clock,  how  many  spaces  has  the  min.  hand 
gained  on  the  hr.  hand  since  5  ?     Draw  a  diagram. 

5.  When  the  hands  are  at  right  angles  between  2  and  3  o'clock, 
how  many  spaces  has  the  minute  hand  gained  since  2?  Draw  a 
diagram. 

6.  At  what  time  between  5  and  6  o'clock  are  the  hands  of  a 
watch  first  20  minute  spaces  apart  ? 

7.  At  what  time  between  2  and  3  o'clock  are  the  hands  of  a 
watch  at  right  angles  ? 

a  Between  4  and  5  o'clock,  when  the  hour  hand  is  as  much 
after  4  as  the  minute  hand  is  before  10,  how  many  minute  spaces 
have  the  hour  and  minute  hands  together  passed  since  4  o'clock  ? 
How  many  spaces  do  they  together  pass  in  1  minute  ? 

9.  In  Ex.  8,  what  is  the  time  ?  Solve  both  with  and  without 
the  use  of  x. 

10.  At  what  time  between  8  and  9  o'clock  are  the  hands  of  a 
watch  together  ? 

11.  A  and  B  start  from  the  same  point  and  travel  in  the  same 
direction.  If  A  travels  6  miles  an  hour,  and  B  4  miles  an  hour, 
how  far  apart  are  they  after  6  hours  ? 

12.  In  how  many  hours  will  A  overtake  B,  if  the  latter  has  5  hr. 
the  start  ? 

la  If  they  travel  in  opposite  directions,  how  far  apart  are  they 
at  the  end  of  6  hours  ? 

14.  If  they  are  60  miles  apart  and  travel  toward  each  other, 
how  far  will  A  travel  before  they  meet  ? 

15.  Two  men,  A  and  B,  26  miles  apart,  set  out  toward  each 
other,  B  30  minutes  after  A;  A  travels  3  mi.  an  hr.,  and  B  4  mi. 
an  hr.     How  far  will  each  have  traveled  when  they  meet  ? 


SOLUTION  OF  PROBLEMS  188 

16.  A  fox  has  60  of  its  leaps  the  start  of  a  hound.  While  the 
fox  makes  5  leaps  the  hound  makes  4 ;  3  leaps  of  the  fox  cover 
the  same  distance  as  2  leaps  of  the  hound.  How  many  leaps 
must  the  hound  make  to  catch  the  fox  ? 

Analysis  Literal  Method 

litlation:  the  bound  must  make  as  Relations :  diHi&nce  hound  mns 

many    leaps   as   the   distance   (in    fox  =  length  of  1  leap  x  no.  of  leaps  ; 

leaps)  he  gains  in  1  leap  is  contained  distance  hound  runs  =  distance 

times  in  the  distance  (in  fox  leaps)  to  fox  runs  +  start  of  fox. 
be  gained. 

Solution:    the  distance   the    bound  Let  5 x  =  no.  ?p. /ox,  300 

goes  in  1  leap  =  the  length  of  I  fox  then  4x  =  no.  Ip.  hound,  240 
leaps,   because   2   leaps  of   the   hound 

cover  the  same  distance  as  3  leaps  of  ^©^  3  a  =  length  1  h.  Ip. 
the  fox.    The  disUnce  the  fox  goes  dur-  ^^en  2  a  =  length  1  /.  Ip. 
ing  the  same  period  is  |  fox  leaps,  be- 
cause the  fox  makes  6  leaps  while  the  12  ax  =  distance  h.  runs 
hound  makes  4.    Therefore,  in  1  leap,  iq  ax  +  120  a  =  distance  h.  runs 
the  hound  gams  (§  —  J)  fox  leaps,  or  ^ 
of  a  fox  leap.  .••  12  ox  =  10  ox  +  120  o 

It  would    take  the  hound  as  many  „  ^^      .o^^ 

1         J.        ■     on.  t      A  1  •  2  ax  =  120  a 

leaps  to  gam  60  fox  leaps,  as  J  is  con- 
tained times  in  60,  or  240  leaps.  x  =  60 

17.  A  fox  pursued  by  a  hound  makes  3  leaps  while  the  hound 
makes  2 ;  but  the  latter  in  3  leaps  goes  as  far  as  the  former  in  7. 
Find  the  length  of  1  hound  leap  in  terms  of  fox  leaps. 

la  Find  the  distance  in  terms  of  fox  leaps  that  the  hound 
gains  in  one  leap. 

19.  If  the  fox  has  60  of  her  own  leaps  the  start,  how  many 
times  will  the  hound  leap  before  he  catches  the  fox  ? 

20.  If  the  fox  has  60  of  the  hound  leaps  the  start,  how  many 
times  will  the  fox  leap  before  she  is  overtaken  ? 

21.  A  hare  is  pursued  by  a  hound.  The  hare  makes  5  leaps 
while  the  hound  makes  3  leaps ;  2  leaps  of  the  Itound  cover  the 
same  distance  as  5  leaps  of  the  hare.  If  the  hare  has  50  of  her 
leaps  the  start,  in  how  many  leaps  will  the  hound  overtake  her  ? 


184  BUYING  AND  SELLING 

BUYING  AND  SELLING 

22.  By  selling  eggs  at  6^  each,  I  shall  lose  24^;  by  selling  at 
10^  each,  I  shall  gain  24^.     How  many  eggs  have  I  ? 

Analysis  Literal  Method 

Belation:  I  have  as  many  eggs  as  Belations:  costof  all  =  1st  sell, 
the  gain  on  1  egg  is  contained  times  in  price  of  all  +  24<^ ;  cost  of  all  = 
the  entire  gain.                                               2d  sell,  price  of  all  -  24  f. 

Solution :  since  the  difference  in  sell-  Let  x  =  no.  eggsy  12 

ing  price  on  1  egg  Is  4^,  the  difference  6x  +  24  =  cost  all 

in  gain  on  1  egg  must  be  4^. '  lOx  -  24  =  cost  all 

Since  I  lose  24*  in  one  case,  and  gain  .-.  6x  +  24  =  10 x  —  24 

24f  in  the  other,  the  difference  in  gain  48  =  4  x 

on  all  the  eggs  is  48^.  x  =  12 

Since  the  difference  in  gain  on  1  egg 

is  4ft  I  must  have  as  many  eggs  to  gain  Proof 

48^,  as  4 f  IB  contained  times  in  48f,  or  1.   12  x    6  +  24  =  96. 

12  eggs.  2.   12  X  10  -  24  =  96. 


2a  If  I  gain  2^  apiece  by  selling  eggs  at  72^  a  dozen,  how 
much  apiece  do  I  gain  by  selling  them  at  60^  a  dozen  ? 

24.  If  I  gain  2^  apiece  by  selling  eggs  at  72^  a  dozen,  how 
much  apiece  do  I  lose  by  selling  them  at  24^  a  dozen  ? 

25.  If  I  sell  eggs  at  96^  a  dozen,  I  gain  on  all  48^  more  than 
if  I  sell  them  at  72^  a  dozen.     How  many  eggs  have  I  ? 

26.  If  I  sell  eggs  at  24^  a  dozen,  I  lose  30^  on  all;  if  I  sell 
them  at  60^  a  dozen,  I  gain  15^  on  all.     How  many  eggs  have  I  ? 

27.  If  I  sell  eggs  at  12^  a  dozen,  I  lose  3^  apiece.  How  much 
a  dozen  must  I  charge  to  gain  3^  apiece  ? 

2a  I  sell  8  eggs  for  a  certain  price.  Had  I  sold  2  more  for 
the  same  money,  the  price  of  each  egg  would  have  been  dimin- 
ished 1^.     For  how  much  did  I  sell  each  egg? 

29.  B  bought  apples  at  2  for  a  cent  and  the  same  number  at  3 
for  a  cent;  he  sold  them  all  at  5  for  2^,  and  thereby  lost  2/. 
How  many  did  he  buy  ? 


SOLUTION  OF  PROBLEMS  185 

LABOR   PROBLEMS 

aa  A  agreed  to  Vork  24  days  for  $  2  a  day  and  his  board,  and 
to  pay  50^  a  day  for  board  when  idle ;  at  the  end  of  the  time  he 
received  $  38.     How  many  days  W^as  he  idle  ? 


Literal  Method 

Relation :  amount  received  for  la* 
bor  —  amount  paid  for  board  =  9  38. 


20  =  5» 

2  =  4 


Analysis 

Relation:  he  was  idle  as  many 
days  as  the  amount  lost  for  each  idle 
day  is  contained  times  in  the  entire  ^^^  ^  =  **^'  ^^'  *^'^'  * 

amount  lost.  then     24  -  x  =  no.  da.  work,  20 

Solution:    on  each  idle  day  he  4S  -  2 x  =  amt.  for  labor 

lost  e  2  that  he  might  have  earned,  ^=amt.for  board 

and  60 />  for  board,  or  $2.60  in  all.  2  * 

If  he  had  worked  the  whole  time,  48  —  2x— -  =  38 

he  would  have  received  $  48  ;  he  lost  o«      a  —  7ft 

through  idleness,  $  48  -  $  38,  or  « 10.  ^     4  x  -  x  -  70 

Since  he  lost  $2.60  for  1  idle  day, 
he  must  have  been  idle  as  many 
days  as  $2.60  is  contained  timee  in  Pboof 

$10,or4day8.  j    20  x  $2  -  4  x  $.60  =  $38. 


31.  A  agrees  to  work  for  40  days  at  $  1.50  a  day  and  his  board, 
and  to  pay  50^  a  day  for  board  when  idle.  How  much  does  he 
lose  each  idle  day  ? 

32.  If  he  receives  $  20  at  the  end  of  the  time,  how  many  days 
was  he  idle? 

33.  A  agrees  to  work  30  days  at  $  3  a  day,  and  to  forfeit  $  1 
a  day  for  every  day  he  is  idle.  How  much  does  he  lose  each 
idle  day  ? 

34.  If  he  receives  $  60  at  the  end  of  the  time,  how  many  days 
did  he  work  ? 

35.  A  agrees  to  work  30  da.  at  $  3  a  day  and  his  board,  and  to 
pay  $1  a  day  for  his  board  when  idle;  at  the  end  of  the  time 
he  receives  $  40.    How  many  days  was  he  idle  ? 


186  INVOLVING   A  PART 

INVOLVING  A  PART 

36.  If  A  can  do  a  piece  of  work  in  6  days,  and  B  in  3  days,  in 
how  many  days  can  they  do  the  work  together  ? 

Relation :  both  can  do  the  work  in  as  many  days,  as  the  part  they  can 
do  in  1  day  is  contained  times  in  the  whole. 

Solution :  in  1  day,  A  can  do  ^  of  it ;  B,  |  of  it ;  both,  the  sum  of  ^  and 
|,  or  y'5  of  it.  It  will  take  them  as  many  days  to  do  the  whole  as  ^  is  con- 
tained times  in  } I,  or  1 1  days. 

37.  A  pastures  5  cows,  and  B,  4  cows.  If  the  whole  expense  is 
$  18,  how  much  should  each  pay  ? 

Relation :  A's  expense  will  be  5  times  the  cost  of  pasturing  1  cow ;  B's 
expense,  4  times  the  cost  of  pasturing  1  cow. 

Since  the  expense  for  0  cows  is  $18,  the  expense  for  1  cow  is  (of  918,  or 
$2.    A's  expense  is  5  x  82,  or  $10;  B's,  4  x  $2,  or  $8. 

3a  A  crew  row  down  stream  8  mi.  an  hr.  and  up  stream  6  mi. 
an  hr.     How  far  down  stream  can  they  row  and  return  in  7  hr.  ? 

Analysis  Literal  Method 

Relation:  they  can  go  as  many  Relation:   no.  hours  down  +  no. 

miles,  as  the  number  of  hours  re-      hours  up  =  7. 
quired  to  go  and  return  1  mile  are  Let  x  =  no.  miles,  24 

contained  times  in  7  hr.  jc 

Solution :    to    row  1   mi.    down  g  =  ^^'  ^^'  ^^^^ 

stream  requires  ^  hr.  ;   up  stream, 

^  hr. ;  both  ways,  ^  +  J.  or  j'^  hr.  r  =  no.  hr.  up 

They  can  go  and  return  as  many  mi. 

in  7  hr.  as  ^f  is  contained  times  in  •*•  I  "*■  I  ~  "^ 

7,  or  24  mi. 


8     6 

3a;  +  4x  =  168 

x  =  24 


39.  A  can  do  a  piece  of  work  in  4  days,  and  B  in  5  days.  VThat 
part  of  the  work  can  A  do  in  1  day  ? 

40.  What  part  can  both  do  in  1  day  ?  How  much  more  can  A 
do  in  1  day  than  B  ?  How  many  days  will  it  take  them  both  to 
do  it? 

41.  C  and  D  together  can  do  a  piece  of  work  in  6  days;  C 
alone^  in  8  days.     How  many  days  will  it  take  D  alone  ? 


SOLUTION   OF   PROBLEMS  187 

42.  A,  B,  and  C  can  do  a  piece  of  work  in  20  days;  A  and  B, 
in  40  days ;  A  and  C,  in  30  days.  In  how  many  days  can  each 
alone  do  it  ? 

43.  Two  pipes  can  fill  a  reservoir  in  8  days ;  with  the  help  of 
a  third  pipe,  they  can  fill  it  in  3  days.  How  many  days  will  it 
take  the  third  alone  to  fill  it? 

44.  A  and  B  can  do  a  piece  of  work  in  6  hr.  After  A  has 
worked  alone  for  3  hr.,  B  commences  and,  working  alone,  finishes 
the  work  in  10^  hr.  In  how  many  hours  can  A  do  the  work 
alone  ? 

45.  A  can  ride  on  a  bicycle  12  miles  an  hour,  and  return  on  the 
cars  30  miles  an  hour.  What  part  of  an  hour  does  it  take  him  to 
ride  1  mile  on  his  bicycle  ? 

46.  How  many  miles  can  he  ride  on  his  bicycle  and  return  by 
the  cars,  in  7  hours  ? 

47.  If  a  steamer  sails  9  mi.  an  hr.  down  stream,  and  5  mi.  an  hr. 
up  stream,  how  far  can  it  sail  down  stream  and  return  in  28  hr.  ? 

48.  A  puts  8  cows  into  a  pasture  for  6  months ;  B,  10  calves 
for  8  months ;  B  pays  $  16.  How  much  should  A  pay,  if  4  cows 
eat  as  much  as  5  calves  ? 

49.  A  and  B  rent  32  A.  of  land  for  $  63.  A  agrees  to  take  12 
A.  of  timber,  and  B,  20  A.  of  meadow  land.  How  much  should 
each  pay,  if  3  A.  of  timber  rent  for  the  same  as  4  A.  of  meadow  ? 

50.  Henry  has  8  marbles,  James  10,  and  Walter  none;  they 
divide  the  marbles  equally  among  them,  and  Walter  pays  6^  to 
Henry  and  James.     How  much  should  each  receive  ? 

51.  A,  B,  and  C  enter  into  partnership.  A  puts  in  $  6000  for 
4  mo.;  B,  $8000  for  3  mo.;  and  C,  $4000  for  6  mo.  If  their 
profits  amount  to  $  5040,  what  is  each  man's  share  ? 

52.  A,  B,  C,  and  D  plant  5464  acres  of  corn.  A  puts  in  1} 
times  as  many  acres  as  B ;  C,  2f  times  as  many ;  and  D,  5^  times 
as  many.  If  they  market  their  crop  for  $  43,712,  what  is  each 
man's  share  of  the  profits  ? 


Letx  = 

:  number 

¥= 

:  tncreote 

6x_ 
8 

:  no.  increased 

6x_ 
3  " 

16 

6z  = 

45 

X  = 

9 

188  A  PART  MODIFIED 

A  PA£T  MODIFIED 
Sa  What  number  increased  by  }  of  itself  becomes  15? 

LlTBRAL   MODIFIKD  LITERAL   MkTHOD 

Relation :   required  uoiuber  +  f  Relation :   required  number  +  } 

of  itself  =  15.  of  itself  =  15. 

Solution :  a  number  increased  by 
§  of  itself  becomes  |  of  itself.  Since 
I  times  the  number  is  15,  the  number 
is  15  -4-  ^,  or  9. 

NoTB.     I  X  No.  »  15;  No.  «  15  •!>  |, 

because  either  of  two  factors  is  equal 
to  their  product  divided  by  the  other. 

Analysis:  since  }  of  the  number 
is  15,  \  of  the  number  is  ^  of  15,  or 
8  ;  },  or  the  number,  is  3  times  3,  or 
9.     See  p.  98,  Ex.  £50,  Note.  Proof.  9  +  J  of  9  =  16 

54.  By  selling  a  watch  for  $40,  a  man  lost  ^.  What  was  the 
cost? 

Business  usage :  the  gain  or  loss  is  always  some  part  of  the  cost. 
Relation :  cost  —  i  of  the  cost  =  selling  price. 

Solution :  the  cost  diminished  by  |  of  itself  becomes  f  of  itself.  Since  | 
times  the  cost  is  $40,  the  cost  is  $40  -^  j,  or  $60. 

55.  An  agent  sold  an  article  for  $  100  on  a  commission  of  ^. 
What  were  the  proceeds  ? 

Business  usage :  when  an  agent  buys,  his  commission  is  some  part  of  the 
buying  price  ;  when  an  agent  sells,  his  commission  is  some  part  of  the  sell- 
ing price. 

Relation :  proceeds  =  selling  price  -  commission. 

Solution :  since  an  agent  sold  on  commission  at  y^,,  his  commission  was  ^ 
of  $  100,  or  $  10  ;  the  proceeds  were  $  100  -  $  10,  or  $  90. 

56.  An  agent  bought  an  article  for  $  100  on  a  commission  of 
■j^.    What  was  the  entire  cost  ? 

Relation :  entire  cost  =  buying  price  +  commission. 
Solution  :  since  an  agent  bought  on  commission  at  j^j,*  ^Js  commission  was 
^  of  $  100,  or  $  10  ;  the  entire  cost  was  $  100  +  §  10,  or  $  1 10. 


SOLUTION  OF  PROBLEMS  189 

57.  What  number  increased  by  5  times  itself  becomes  30  ?  by  4 
times  itself,  20? 

sa  What  number  increased  by  J  of  itself  becomes  20  ?  by  ^  of 
itself,  30  ? 

59.  What  number  diminished  by  J  of  itself  becomes  30  ?  by  | 
of  itself,  20? 

60.  A  horse  cost  $  60  and  was  sold  at  a  gain  of  \.  What  was 
the  selling  price  ? 

61.  A  horse  cost  $  60  and  was  sold  at  a  loss  of  J.  What  was 
the  selling  price  ? 

62.  By  selling  a  horse  for  $  40, 1  lose  \.    How  much  did  he  cost  ? 

63.  By  selling  a  horse  for  $  36,  I  gain  ^.  How  much  did  he 
cost? 

64.  By  selling  a  horse  for  $  40,  I  gain  \.  By  how  much  must 
I  increase  my  price  to  gain  ^? 

65.  If  my  gain  was  j,  or  $40,  what  was  the  selling  price  ?  the 
cost  ? 

66.  If  my  loss  was  ^,  or  $20,  what  was  the  cost?  the  selling 
price  ? 

67.  If  I  buy  at  $3  and  sell  at  $4,  what  part  do  I  gain? 

ea  If  I  sell  J  of  an  article  for  what  the  whole  cost,  what  part 
of  the  whole  do  I  gain  ? 

69.  An  article  was  sold  at  a  gain  of  -fj^ ;  if  it  had  cost  $  120 
more,  the  same  selling  price  would  have  entailed  a  loss  of  -^. 
Find  the  cost. 

7a  A  man  sold  an  article  for  40^,  and  thereby  gained  J  as 
much  as  if  he  had  sold  it  for  60^.     What  was  the  cost? 

71.  An  agent  sold  flour  for  $200  at  a  commission  of  ^.  What 
was  the  commission  ?  the  proceeds  ? 

72.  An  agent  bought  flour  for  $200  at  a  commission  of  -^. 
What  was  the  commission ?  the  entire  cost  to  his  employer? 

7a  An  agent  sold  flour  at  a  commission  of  ^,  and  received  as 
commission,  $25.    What  were  the  proceeds  ? 


pp:rcentage 


TERMS   AND   RELATIONS 

In  many  cases,  it  has  become  custom- 
ary to  use  the  term  per  cent,  written  %, 
in  place  of  hundredths. 

The  number  on  which  the  per  cent 
is  computed  is  the  base;  the  product  is 
the  percentage;  the  base  -f  the  percent- 
age, the  amount;  the  base  —  the  per- 
centage, the  difference. 

%  means  hundredths.     See  p,  114* 

A  per  cent  expression  may  be  changed  to  a  fraction  or  an 
integer,  and  conversely. 


Illustration 

.06  of  200  =  12 

Or, 
6%  of  200  =  12 

200,  base 
12,  percentage 
212,  amount 
188,  difference 
ejo  means  .06 


5.  .16  to  per  cent 

6.  .33J  to  per  cent. 

7.  ^  to  per  cent, 
a  4  to  per  cent. 


Reduce  : 

1.  16%  to  a  decimal. 

2.  33 1  %  to  a  decimal. 
a  33  J  %  to  a  fraction. 
4.  400%  to  an  integer. 

.33^  =  \. 

9.   State  rapidly  the  per  cent  equivalents :  f ,  |,  f ,  f ,  |^,  J,  J,  J, 

TVif^iiii*»TV»f     Seep.m. 

10.   State   rapidly  the   fractional   equivalents:    62J%,   87^%, 
37.1%,  33.^%,  16|%,  2^%.  40%,  75%,  60%,  20%,  80%,  8^%, 
fo,  12^-%,  50%.     See  p.  IIJ,. 

190 


Ex.  a  33}% 
Ex.  4.  400% 


Ex.  7.   \ 
Ex.  a  4 


.331=331% 

m  =  4oo% 


PERCENTAGE  191 

CASE  I  — DIRECT 

U.  What  is  6%  of  60  ?  12.  What  is  16 J%  of  486  ? 

50  486 

.06  .16^ 

3.00  "TT 

6%  of  60  is  .06  of  50,  or  3.  16|%  of  486  is  J  of  486,  or  81. 

Note.  —  A  per  cent  expression  must  be  reduced  to  a  decimal  or  to  a  common 
fraction,  before  it  can  be  used  as  a  multiplier. 

13.  A  farmer  having  360  sheep,  lost  5%.     How  many  had  he 

left? 

S60  =  no.  sheep 
Qg  He  lost  .05  of  360  sheep,  or  18 

sheep ;  he  had  left  the  difference,  or 


18.00  =  no.  lost  342  sheep. 

S42  =  wo.  left 

14.  What  is  16%  of  5250  ?    18%  of  3825  ?     66f  %  of  3483  ? 

15.  A  man  wills  17%  of  his  property  to  his  son,  and  18%  to 
his  daughter.  If  his  property  is  worth  $  8200,  how  much  will 
each  receive? 

16.  A  has  $625;  B  has  86%  as  much  as  A;  C,  78%  as  much 
as  B.     How  much  has  C  ? 

17.  A  lady  buys  55  yards  of  muslin  at  8^  a  yard,  and  27  yards 
of  cloth  at  65^  a  yard.  If  she  pays  66%  of  the  bill,  b<>\v  unieh 
does  she  still  owe  ? 

la  A  man's  salary  is  $75  a  month;  if  he  spends  65%  ot  his 
salary  each  month,  how  much  will  he  have  at  the  end  of  6 
months  ? 

19.  A  gentleman  gave  $  11.20  to  his  children ;  his  sons  received 
65%  of  the  money,  and  his  four  daughters  the  remainder.  How 
much  did  each  daughter  receive  ? 

20.  A  man  has  a  library  of  1600  volumes ;  14%  are  biography ; 
62%,  history,  and  83^%  of  the  remainder,  fiction.  How  many 
volumes  of  fiction  are  there  in  his  library  ? 


192 


INDIRECT  CASES 


CASE   II  — INDIRECT 


21. 

ber? 


3  is  6%  of  what  num- 

.06 

The  equation  is  readily  formed  by 
substituting  '  =  *  for  'is*;  'x'  for 
*of»'  and  *No.'  for  'what  number.* 

Since  either  factor  is  equal  to  the 
product  divided  by  the  other  factor, 
No.  =  3  ^  .00,  or  60. 

Note.  —  A  per  cent  expressioD  must  b«  reduced  to  a  decimal  or  to  a  common 
fraction,  before  it  can  be  used  as  a  divisor. 

23.  A  farmer  lost  18  sheep,  or  5%  of  his  flock.     How  many 

had  he  at  first? 


22.  Of  what  number  is  81, 
16i%? 

161%  ^o.=81 

No.  =  81-t--  =  486 

The  sign  of  multiplication  may  be 
omitted. 

Since  either  factor  is  equal  to  the 
product  divided  by  the  other  factor, 
No.  =  81  -i-  i,  or  486. 


5%  F=  no.  lost 
18  =  no.  lost 
.-.  5%F=^18 


360 


Of  what  number: 

24.  Is  12,  50%? 

25.  Is  368,  23%? 

26.  Is  57,  15%? 

27.  Is  522,  18%? 


Proof 

360  =  no.  at  first 
.06 

18  =  no.  lost 


2a  Is  10.5,  12i%? 

29.  Is  40.5,  7 J %? 

3a  Is  4578,  84%? 

31.  Is  7735,  85%? 


32.  What  is  a  man's  income  if  31 J  %  is  ^600  ? 

33.  A  general  lost  16  J  %  of  his  army;  315  killed,  110  prisoners, 
and  70  deserters.     How  many  men  had  he  at  first  ? 

34.  A  man  paid  $750  for  a  house,  which  was  24%  of  what  he 
paid  for  160  A.  of  land.    What  was  the  cost  of  the  land  per  acre  ? 

35.  Mr.  Turner  earns  $85  a  month;  his  salary  for  the  year  is 
68%  of  his  brother's  salary  for  8  months.  How  much  does  his 
brother  receive  a  month  ? 


»1  = 

n 

x4SG 

R- 

81 

= 

i  =  ie\f. 

The  \ 

must 

be 

reduced  to  %  be- 

cause 

the 

1  answer 

is  to  be  expressed 

in  0/,. 

PERCENTAGE  193 

CASE   III  — INDIRECT 
36.  3  is  what  %  of  60  ?  37.  81  is  what  %  of  486  ? 

3  =  Ex  50 

The  equation  is  readily  formed  by 
substituting  *  = '  for  '  is '  ;  •  K,'  for  %, 
and  'x'  for  'of.' 

38.  A  farmer  having  360  sheep,  lost  18.     What  %  did  lie  lose  ? 

18=  Rx  360 

^  -         y  lielation :  the  number  lost  is  some 

R  =  ^^  =  —  z=5%  %  of  the  whole  number. 

360     20        ' 

What  % : 

39.  Of  45  is  15?  42.  Of  1311  is  437? 

40.  Of  72  is  18  ?  43.  Of  2288  is  286  ? 

41.  Of  78  is  63?  44.  Of  2700  is  300? 

45.  A  man  worth  $250,000  lost  $27,500.     What  %  remained? 

46.  Mr.  Brown  works  6  days  each  week  for  44  weeks  of  the 
year.  What  per  cent  of  the  time  is  he  idle,  counting  300  working 
days  to  the  year  ? 

47.  The  population  of  a  certain  town  is  57,500 ;  of  these,  3450 
are  Irish;  22,540,  Spanish ;  and  the  remainder,  English.  What 
per  cent  of  the  population  is  English  ? 

4a  A  man  having  $  1275,  spent  $210  for  a  carriage,  $112  for 
a  horse,  and  $86  for  a  harness.  What  per  cent  of  his  money 
remained  ? 

49.  A  rectangular  field  75  feet  long,  is  J^  as  wide.  The  breadth 
is  what  per  cent  of  the  length  ? 

50.  The  product  of  two  numbers  is  11,250;  the  first  is  126. 
What  per  cent  of  the  first  is  the  second? 

AMER.    ARITH. — 13 


194 


INDIRECT  CASES 


CASE  IV  — INDIRECT 

51.   What  number  increased  by  6%  of  itself,  becomes  53  ? 
100%  X=theno. 

6%  N=the  increcue  Proof 

60  =  the  no. 
,06 
3  =  the  increase 


fo  N=th€  no.  incr. 
53  =  the  no.  incr, 
106%N=53 


1.06 


63  =  the  no.  incr. 


52.  What  number  diminished  by  16|  %  of  itself,  becomes  405  ? 
100%  N=  the  no. 


16\%  N=  the  decrease 
8S\%  N=  the  no.  deer. 
405=  the  no.  deer. 
,\  83l%N=405 

N =405+1=^486 


Pboof 
486  =  the  no. 
.83^ 
81  =  the  decrease 
406  =  the  no.  deer. 


53.   After  losing  5%  of  his  sheep,  a  farmer  had  342  left.     How 
many  had  he  at  first  ? 

100%F=no.  at  first 
5%  F=no.  lost 
95%F=no.  left 
342  =  no.  left 
.-.  95%F=S42 

F=^  =  360 
.95 

WhaJt  number  increased  by : 

54.  66f  %  of  itself  becomes  30  ? 

55.  6%  of  itself  becomes  212  ? 

56.  25%  of  itself  becomes  60  ? 


Proof 
360  =  no.  at  first 

.05 

18  =  no.  lost 
342  =  no.  left 


What  number  diminished  by : 

57.  25%  of  itself  becomes  30  ? 

58.  6%  of  itself  becomes  188  ? 

59.  10%  of  itself  becomes  81? 


PERCENTAGE  195 

MISCELLANEOUS   PROBLEMS 

6a  After  16%  of  a  heap  of  grain  was  taken  away,  there 
remained  252  bushels.     How  many  bushels  were  there  at  first? 

d.  In  1890,  the  population  of  a  town  was  17,280,  which  was 
35%  more  than  in  1880.     What  was  the  population  in  1880  ? 

62.  My  crop  of  corn  this  year  is  12^  %  greater  than  last  year, 
and  I  have  raised,  during  the  two  years,  3825  bushels.  What  was 
my  last  year's  crop  ? 

63.  A  young  man  having  received  a  fortune,  deposited  60%  of 
it  in  bank ;  he  afterward  drew  30%  of  his  deposit,  and  then  had 
$  7560  in  bank.     What  was  his  entire  fortune  ? 

64.  Mr.  Black's  property  is  valued  at  $  15,000,  and  85%  of  his 
property  is  2%  more  than  his  brother's  property.  What  is  the 
value  of  his  brother's  property  ? 

65.  At  a  forced  sale,  a  bankrupt  sold  his  farm  for  $  6642,  which 
was  18%  less  than  its  real  value.   What  was  the  value  of  the  farm  ? 

66.  Into  a  vessel  containing  pure  vinegar,  there  were  poured 
12^  gallons  of  water,  which  was  16|%  of  the  mixture.  What 
was  the  quantity  of  pure  vinegar  ? 

67.  A  man  paid  $  16.20  for  the  use  of  land  which  cost  $  360. 
What  %  did  the  owner  realize  on  his  investment  ? 

68.  From  a  cask  containing  24  gal.  3  qt.,  17  gal.  3f  qt.  leaked 
out     What  %  leaked  out  ? 

69.  A  fruit  grower,  having  sent  2200  baskets  of  peaches  to 
Philadelphia,  found  that  11%  of  them  had  decayed.  If  he  sold 
the  balance  at  76^  a  basket,  what  sum  did  he  receive  for  his 
peaches  ? 

70.  On  Jan.  1,  a  man  weighed  160  lb.  In  January  he  lost  2^% 
in  weight,  and  in  February  gained  2j%.  What  %  of  his  weight 
Jan.  1  was  his  weight  on  the  first  day  of  March  ? 

71.  How  many  gallons  of  water  must  be  mixed  with  70^  gallons 
of  wine,  so  that  the  mixture  may  contain  6%  of  water  ? 


196  PROFIT  AND  LOSS 

PROFIT  AND  LOSS 

In  buying  or  selling,  the  gain  or  loss  is  If  I  buy  for  $  100  and 

always  some  %  of  the  cost.  »«"  ^^r  « 110,  the  gain  is 

^               '      "^  «10,  or  10%  of  the  cost. 
This  principle  is  established  by  business  usage. 

72.  A  boat  was  bought  for  $  9136.50,  and  sold  at  a  loss  of  3%. 
What  was  the  selling  price  ? 

f  9136.50  =  cost  Note.  — Since  our  smallest  coin  is 

Q^  one  cent^  we  give  the  answer  to  the 

nearest  cent.     The  loss  is  $274.10. 

f  274-095  =  lo88  Count  5  or  more  mills  as  1  ^  ;  neglect 

9136.50  less  than  5  mUls. 


$886240    =  selling  price 

73.  The  profit  on  the  sale  of  a  horse  was  $39.20,  or  14%. 
What  was  the  cost? 

14%  C:^  gain  Proof 

f  39.20  =  gain  1280  =cost 

.-.  14%  C=  39.20  ^ 

n_S9.20_^gQ  1120 

^"TTT"  280 


Cost  =  $280.  $  39.20  =  gain 

74.   A  house  was  sold  for  $320.32  at  a  gain  of  12%.    What 
was  the  cost  ? 

100%  C  =  cost 

12%  C  =  gain  Proof 

112%  C  =  selling  pnce  '     ^  286  =  cost 

320.32  =  selling  price 
.'.  112%  C  =  320.32  J^/^'^  =  ga^° 

0  = 


2J2  ♦  320.32  =  selling  price 

Cost  =  286. 


PERCENTAGE  197 

75.  I  sold  a  plow  for  $  16.32,  and  thereby  lost  4%.  What  was 
the  cost  ? 

76.  My  sales  exceeded  the  cost  by  $  240 ;  the  sales  were  $  560. 
What  was  the  gain  %  ? 

77.  I  sold  silks  for  $270.00,  thereby  losing  10%.  What  sell- 
ing price  would  have  made  the  gain  $  26.50  ? 

7a  A  merchant  bought  540  yards  of  muslin  at  7  f,  and  sold  it 
at  a  reduction  of  2^%.     What  was  the  entire  loss  ? 

79.  I  bought  a  quantity  of  wheat  at  75^  a  bushel.  At  what 
price  must  it  be  sold  to  gain  16%? 

80.  I  bought  75  horses  at  $  5(j  each,  expenses,  $S.  If  I  lost  8 
of  them,  at  what  average  price  must  I  sell  them  to  gain  15%? 

81.  By  selling  for  $2000,  I  gained  14%  of  the  selling  price. 
Had  I  gained  14%  of  the  cost,  how  much  would  1  have  received? 

82.  How  shall  I  mark  shoes  that  cost  $2.50,  so  that  I  may 
deduct  20%  from  the  marked  price  and  still  make  10%? 

83.  I  sold  a  carriage  at  20%  gain,  and  with  the  money  bought 
a  horse  which  I  sold  for  $168.30,  thereby  losing  15%  of  the 
selling  price  of  the  carriage.     What  was  the  cost  of  the  carriage? 

84.  I  began  business  with  $24,000;  gained  16%  the  first  year 
and  added  it  to  the  capital.  What  was  my  capital  at  the  begin- 
ning of  the  second  year  ? 

85.  4%  of  my  capital  was  invested  in  18  watches  at  $180  a 
dozen;  7%  of  my  capital  was  invested  in  clocks  at  $2.10  each. 
How  many  clocks  had  I  ? 

86.  A  merchant  marked  goods  at  16  J  %  above  cost,  and  sold 
the  goods  at  this  marked  price  for  $  350,  What  was  the  cost  ? 
What  %  would  he  have  gained  by  selling  the  goods  for  $375? 

87.  A  man  gained  33^%  on  the  sale  of  a  horse,  20%  on  the 
sale  of  a  carriage,  and  50%  on  the  sale  of  a  buggy.  If  the  selling 
price  of  each  was  $  150,  what  was  his  entire  gain  ? 

88  A  man  lost  10%  of  his  money,  then  gained  10%  of  what 
he  had  left,  and  then  had  $396.     How  much  had  he  at  first  ? 


198  COM^IISSION 

COMMISSION 

A  person  may  be  employed  A  farmer  takes  to  a  grocer  10  bar- 
to  buy  or  sell  for  another.     The  ^*»  ^^  apples  to  be  sold  at  1 2  a  barrel, 
employer  is  the  principal,  the  »«^»ng  ^  Pay  1^%  commission  for 
1         ,    ,,               ^     XL         •  selling  them.      It  is  just  that  the 
employed,  the  agent;  the  price  ^^^^  ^^^^^^  ^^^p  ,,,^  ^^  ^^^  ^^^^ 

paid  for  the  service,  the  com-  a  manufacturer  sends  his  agent 

mission ;    the  amount  returned  %  1060  to  buy  wool,  after  deducting 

to  the  principal,  the  nef  proceeds.  »  commiasion  of  6%.    If  the  agent 

„               ,,  takes  6%  of  $1050,  he  will  take  6% 

Business  Usaob  *     u  /  u              *      .,            ,        f 

of  what  he  pays  for  the  wool,  and 

If  an  agent  aelUy  his  commis-  also  6%  of  what  he  keeps.    It  is  not 

sion  is  some  per  cent  of  the  sales,  just  that  he  should  receive  5  %  of 

If  an  agent  buys,  his  commit-  ^*»a^  *»«  keeps,  because  he  performs 

sion  is  s(me  per  cent  of  tiie  pnr-  ''^  ^^^^  ^°'  ^^'  ^"^  ^^  ^  ^"^^•'^^  ^ 

,  6  %  of  what  he  pays  for  the  wool. 

89.  Find  the  amount  of  sales,  when  an  agent  receives  $  40  at  a 
commission  of  20%. 

20%  S  =  40 

jg  __  ^^  _  £QQ  Since  the  agent  sells,  his  commis- 

.20  sion  is  20  %  of  sales. 
Sales  =  f  200. 

90.  Find  the  amount  of  sales,  when  the  principal  receives  $  40 ; 
agent's  commission  20%. 

100%  S  =  sales 

20%  S  =  com.  Proof 

80%  S  =  proceeds  f  50  =  sales 

f  40  =  proceeds  j^ 

foS  =  40  10  =  com. 

,Q  j60 

^^~sd^^^  ♦40  =  proceeds 
Sales=f50. 


PERCENTAGE  199 

91.  At  3%,  what  is  an  agent's  commission  on  $  1150  sales  ? 

92.  At  3%,  what  is  an  agent's  commission  on  $  1150  purchase  ? 

93.  Find  the  commission  when  an  agent  receives  $259.60  to 
be  invested  in  goods,  after  deducting  his  commission  of  10%. 

94.  How  many  pounds  of  sugar  at  8^  a  pound  can  an  agent 
buy  for  $  15.99,  after  deducting  his  commission  of  2^%  ? 

95.  Find  the  rate  per  cent  of  commission  when  $  2.40  is  paid 
for  a  sale  of  $  160. 

96.  Find  the  amount  of  sales  when  a  commission  of  2\fo  pays 
the  agent  $  6.48. 

97.  Find  the  net  proceeds  from  the  sale  of  36  barrels  of  sugar 
at  $4,  commission  6J%. 

98.  Find  the  amount  of  the  purchase  when  an  agent  receives 
i  351  to  invest  in  sugar  after  deducting  his  commission  of  8^. 

99.  My  principal  instructed  me  to  invest  $  1220  in  wool,  and 
sent  me  a  draft  for  $  1220  plus  2%  commission.  What  was  the 
amount  of  the  draft  ? 

100.  My  principal  instructed  me  to  invest  $1244.40  in  wool 
after  deducting  my  commission  of  2%.  What  was  my  commis- 
sion? 

101.  After  selling  wheat,  an  agent  deducts  $56  commission, 
and  sends  his  principal  $  2744.  What  rate  of  commission  does 
he  receive  ? 

>02.  An  agent  bought  4000  bushels  of  oats ;  his  commission  at 
2Xfo  was  $  28 ;  charges  for  freight  and  storage,  $  5'2.  How  much 
per  bushel  did  the  oats  cost  the  principal  ? 

103.  I  sold  on  a  commission  of  4^%,  320  barrels  of  sugar  at 
$  18,  and  60  barrels  of  oil,  44  gallons  to  the  barrel,  at  15^  a  gallon. 
How  much  did  I  remit  to  my  employer  ? 

10*.  An  agent  sells  1200  barrels  of  apples  at  $4.50  a  barrel, 
and  charges  2^%  commission.  After  deducting  his  commission 
of  8%  for  buying,  he  invests  the  net  proceeds  in  flour  at  $5  a 
barrel.     What  is  his  entire  commission  ? 


200  TAXES 

TAXES 

In  some  states,  every  male  citizen  is  annually  taxed  a  small 
amount,  jwll  tax,  without  regard  to  his  property ;  property  owners 
pay  an  additional  tax.  In  these  states,  the  entire  tax  is  dimin- 
ished by  the  sum  of  the  poll  tax,  and  the  remainder  is  divided  by 
the  assessed  value  of  all  the  taxable  property,  to  find  the  tax  on 
each  dollar. 

Tfie  collector  usually  receives  some  %  of  the  amount  collected. 

105.  A  tax  is  $16,020;  the  taxable  property  $784,760;  the 
number  of  polls  at  $1.25  is  260.  A*s  property  is  $  7800;  what  is 
his  tax  ? 

f  1.25  f  7800  =  As  property 

eeo  .02. 

fS25  =  poll  tax  $156.00  =  As  property  tax 
_16fi20  =  whole  tax  ^ ^^  ^  ^,^  ^^^  ^^^ 

f  15f695  =  property  tax  

784750 )15695.00(. 02  f  i57.^5  =  A's  whole  tax 
15695.00 

106.  The  tax  on  A*s  property  at  4  mills  on  a  dollar  is  $  615.80. 
What  is  the  assessed  value  of  his  property  ? 

107.  The  school  tax  of  a  certain  town  is  $  4782.  If  the  rate  of 
taxation  is  3  mills  on  a  dollar,  what  is  the  amount  of  taxable 
property  ? 

loa  A  town  is  to  be  taxed  $  13651.48 ;  the  taxable  property  is 
$865,432;  the  number  of  polls  at  $1  is  670.  What  is  B's  tax, 
whose  property  is  assessed  at  $9720? 

109.  The  whole  amount  of  taxable  property  in  a  certain  town  is 
$  386,722 ;  there  are  also  1560  polls  at  $  1.50  each.  If  the  rate  of 
taxation  is  5  mills  on  a  dollar,  what  is  the  tax  ? 

110.  A  tax  of  $  14,846  is  to  be  assessed  on  a  certain  village ; 
the  property  is  valued  at  $  1,060,000,  and  there  are  2123  polls  at 
$  2.  What  is  the  assessment  on  a  dollar  ?  What  is  Mr.  Doan's 
tax,  whose  property  is  assessed  at  $  11,600  ? 


PERCENTAGE  201 

TRADE  DISCOUNT 

Wholesale  dealers  and  manufacturers  issue  price  lists  of  their 
goods.  As  the  market  varies,  they  change  the  rate  of  discount 
instead  of  changing  the  fixed  price.  They  frequently  offer  other 
discounts  and  an  additional  discount  for  cash. 

111.  A  man  sold  a  bill  of  goods,  list  price  $  100,  on  4  mo.  at 
.'),  10,  and  6%  off  for  cash.    What  was  the  net  price  for  cash  ? 

^100  =  list 

£  —  ^^^  ^*^*  Meaning :  If  the  goods  are  paid  for  at  the 

^95  =  after  1st  end  of  4  mo.,  5%  will  be  deducted  from  the  list 

g  r/i  _  Qnfi  fiio  price,  and  10%  from  the  remainder.     If  cash  is 

! —  ~  *  paid  at  the  time  of  purchase,  an  additional 

$85.50  =  after  2nd  discount  of  6%  will  be  deducted  from  the  last 

5.13  =  cash  dis.  remainder. 


$80.37=  net  price 

112.  A  man  sold  a  bill  of  goods,  list  price  $  100,  on  4  mo.  at  10, 
0,  and  6%  off  for  cash.     What  was  the  net  price  for  cash  ? 

113.  If  the  purchaser  decides  not  to  pay  cash,  is  one  of  the 
above  offers  better  for  him  than  the  other  ? 

114.  A  merchant  offered  a  bill  of  goods,  list  price  $100,  on 
3  mo.  at  8,  and  4%  off  for  cash.     What  was  the  cash  price  ? 

115.  A  merchant  offered  a  bill  of  goods,  list  price  $100,  on 
3  mo.  at  4%,  and  8%  off  for  cash.     What  was  the  cash  price  ? 

116.  If  the  purchaser  had  decided  not  to  pay  cash,  which  of 
the  offers  would  have  been  to  his  advantage ;  that  in  Ex.  114,  or 
that  in  Ex.  115  ? 

117.  A  bill  of  goods,  list  price  $100,  was  offered  at  45%  dis- 
count; or  at  30%  and  15%  off.  Which  offer  was  better  for  the 
purchaser  ? 

lia  Which  is  better  for  the  purchaser,  55%  discount,  or  two 
successive  discounts  of  50%  and  10%? 


202  INSURANCE 

INSURAJNCE 

An  agreement  to  pay  for  loss  or  damage  is  insurance.  The 
indemnity  may  be  against  loss  on  i^vo^rty,  property  insurance; 
by  fire,^re  insurance;  or  by  accidents  of  navigation,  marine  insur- 
ance. The  indemnity  may  also  be  against  personal  injury,  or  loss 
of  life,  life  insurance. 

The  contract  or  agreement  between  the  insurance  company  and 
the  person  protected  is  the  policy;  the  sum  paid  for  insurance, 
the  pretnium. 

The  premium  is  some  per  cent  of  the  amx)unt  insured. 

119.  A  store  worth  $  20,000  was  insured  for  ^  of  its  value  at 
1\%'     What  was  the  premium  ? 

$16000  =amt.  insured 

Q2 1  Relation :  the  premium  is  some  %  of 


the  amount  insured. 


$  200.00  =  premium 

120.  Find  the  premium  to  be  paid  for  insuring  a  person's  life 
for  $  5000,  at  an  age  for  which  the  rate  is  2J%. 

121.  For  what  sum  should  a  cargo  worth  $  60,000  be  insured  at 
4%,  so  that,  in  the  event  of  loss,  the  owner  may  receive  both  the 
value  of  the  cargo  and  the  premium  ? 

122.  A  man  paid  $125  for  insuring  at  |%.  What  was  the 
amount  insured? 

123.  I  paid  $  53,  including  cost  of  policy,  $  1.75,  for  insuring 
my  property  to  the  amount  of  $  8200.     What  was  the  rate  ? 

124.  What  is  the  amount  of  the  annual  premium  at  $  35.20  per 
3 1000,  on  a  life  policy  of  $  6500  ? 

125.  I  paid  $40.50,  at  1^%,  for  insuring  my  house  for  |  its 
value.     What  is  the  value  of  the  house  ? 

126.  A  man  35  years  old,  takes  out  a  $6000  life  insurance 
policy  payable  when  he  reaches  the  age  of  b5.  If  he  pays  an 
annual  premium  of  $  63.10  per  $  1000,  how  much  money  will  he 
have  paid  in  if  he  lives  till  the  policy  falls  due  ? 


PERCENTAGE  203 

DIFFICULT  PROBLEMS 

Care  should  be  used  in  deciding  what  the  *%  is  of.' 

127.   I  mix  20%  of  rosin  with  8  lb.  of  tallow.     How  many 

pounds  of  rosin  in  the  mixture  ? 

100%  .yf  =  mixture 
S0%  M  =  rosin 


80%  M=  tallow  The  20%  is  of  the  mixture  and  not  of  the 

S  lb.  =  tallow  rosin ;  another  form  of  statement  would  be 

•'•  80%M-8  *'20%  of  a  mixture  is  rosin." 

.80 
Ans.  S  lb.  rosin. 

Generally,  it  is  best  to  represent  what  the  *%  is  of  by  100% 
of  that  thing. 

12a  A  quantity  of  sugar  was  sold  at  10  per  cent  gain.  If  it 
had  cost  $  120  more,  the  same  selling  price  would  have  entailed  a 
loss  of  10  per  cent.     Find  the  cost  of  the  sugar. 

100%  C  =  cost  100 %  C  =  cost 

100%  C+  120  =  sup.  cost  10%  C  =  gain 

10%C-\-    IS  =  sup.  loss  1lO%C  =  sell p, 
90%  C+  108=  sell.  p. 

.'.  110%C=90%C+108 
110%  C-90%  C=108 
20%  C=  108 

€='-^  =  640 
.20 

Cost  =  f  540. 
Sometimes  it  is  more  convenient  to  represent  what  the  '%  is 
of '  by  a  number. 

129.  I  mix  20%  of  rosin  and  80%  of  tallow.  AVhat  per  cent 
of  the  weight  of  the  tallow,  is  the  weight  of  the  rosin  ? 

Let  6  lb.  =  wt,  mixture  Since  rosin  is  some  %  of  tallow^ 

jl^L.  l  =  Rx4 

J  lb.  =  t€t.  rotin  R  =  lz^25H. 

4  lb.  =  u>t.  tallow  4 


204  DIFFICULT  PROBLEMS 

In  some  problems,  there  is  %  of  two  different  bases,  and 
neither  base  is  given. 

130.  How  many  per  cent  above  cost  must  a  man  mark  his 
goods,  in  order  to  take  off  10  per  cent  and  still  make  a  profit 
of  12^%? 

100  %C  =  cost  200%  M=  marked  p. 

li^-%  C  =  gain  10%  M=  deduction 

IJil^C^sellp.  90%M  =  seilp. 

.-.  90%M=llg{%C 

Mark, p,i»t5% above  cost. 

In  some  problems,  two  equations  with  two  unknown  quantities 
axe  necessary. 

131.  My  agent  sold  my  flour  at  4%  commission.  Increasing 
the  proceeds  by  $168,  I  bought  wheat,  paying  2%  commission; 
wheat  declining  3%,  ray  loss,  including  commissions,  was  $30. 
AVhat  was  the  selling  price  of  the  flour  ? 

100  %  S  =  selUng  flour  100  %  C  =  cost  teheat 

4%S  =  com.  S%C  =  com. 


96%  S  =  proceeds  102 %  C  =  invst.  wheat 

96%  S  -\- 168  =  invst.  wheat  5%C  =  dec.  and  com. 


4%S^5%C=S0  (1) 

96%  S-  102 %  C  =  -  168     (2) 


Since    the    loss,    including    com- 
missions, was  §.30,  we  obtain  (1). 
96%S-\-  120%C—      720     (3)  Since    the    investment   in  wheat 

96% S-  102%C  =  -  168     (2)      was  $168  more  than  the  proceeds 


222^  C  —      888     r4'i       of  the  flour,  we  obtain  (2). 


C  =  400 


We  solve  these  equations  as  on 

COH  +  Com.  Z  UOS.  P, '«'  ■•  •'\^,  (>)  ^''^  (•"'^  ■'  (^)  '"■" 

Sell.  p.  Flour  =  %iSO.  (3)  gives  (4)  ;  .  .  .  . 


PERCENTAGE  205 

132.  What  per  cent  would  a  dishonest  dealer  gain  by  using  a 
false  weight  of  15  oz.  instead  of  a  pound  ? 

133.  What  per  cent  does  a  customer  lose,  if  his  grocer  uses  a 
false  weight  of  15  oz.  instead  of  a  pound  ? 

134.  The  lead  ore  from  a  certain  mine  yields  40%  of  metal,  and 
of  the  metal,  |%  is  silver.  How  many  ounces  of  silver  will  be 
obtained  from  a  ton  of  ore  ? 

135.  A  brewery  is  worth  4%  less  tlian  a  tannery,  and  the  tan- 
nery 16%  more  than  a  boat;  the  owner  of  the  boat  has  traded  it 
for  75%  of  the  brewery,  thus  losing  $103.  How  much  is  the 
tannery  worth  ? 

136.  8  lb.  of  a  certain  article  loses  3  oz.  in  weight  by  drying. 
What  per  cent  of  the  original  weight  is  water? 

137.  A  man  sold  an  article  at  20%  gain ;  had  it  cost  f  300  more, 
he  would  have  lost  20%.     What  was  the  cost  ? 

13a  How  must  I  mark  goods  so  that  I  may  deduct  10%  from 
the  marked  price  and  still  make  17%  ? 

139.  A  merchant  marked  damaged  goods  at  a  certain  per  cent 
below  cost,  but  sold  them  for  cost  at  an  advance  of  11 J  %  on  the 
marked  price.     What  was  the  marked  price  ? 

140.  An  agent  sold  my  corn,  and,  after  reserving  his  commis- 
sion, invested  all  the  proceeds  in  corn  at  the  same  price;  his  com- 
mission, buying  and  selling,  was  3%  each,  and  his  whole  charge 
$  12.     For  how  much  was  the  corn  sold  ? 

141.  A  dealer  sold  wheat,  losing  4%  ;  keeping  $  18  of  the  pro- 
ceeds, he  gave  the  remainder  to  an  agent  to  buy  corn,  8%  com- 
mission; his  loss  together  with  the  commission  was  $32.  How 
much  did  the  agent  pay  for  the  corn? 


STOCKS   AND   BONDS 


TERMS  AND   RELATIONS 


For  the  prosecution  of  a  business 
enterprise,  individuals  sometimes 
form  a  stock  company.  Officers  are 
elected,  and  a  charter  (authority  to 
do  business  as  an  individual)  is  ob- 
tained from  the  state. 

The  engraved  part  of  a  certificate 
states  the  number  of  parts  (shares) 
into  which  the  property  (stock)  is 
divided,  the  face  value  (par  value) 
of  each  part,  etc.  The  written  part 
states  the  number  of  shares  bought, 
the  name  of  the  purchaser,  etc. 

206 


From  a  study  of  the  above 
certificate,  the  pupil  should  an- 
swer the  following  questions : 

Where  is  the  Altar  mine 
situated  ? 

Who  was  the  president  of 
the  Altar  Mining  Co.?  the 
secretary  ? 

Into  how  many  shares  is 
the  property  divided  ? 

What  is  the  par  value  of  1 
share  ? 

What  is  the  capital  stock  ? 

Who  is  the  owner  of  this 
certificate  ? 

How  can  this  certificate  be 
transferred? 


STOCKS  AND  BONDS  207 

Usually,  shares  do  not  sell 

for  their  par  value,  but  for  less  John  Fluker  bought  the  stock  when 

i;it  a  discount),  or  for  more  (at      »^  ^^  *^7^  '««"«'^'  ^^«2  a  share,  or 
^  .       ^    ^'         ,.  \  at  2%  of  the  par  value,  or  at  98% 

a  premium),  according   to  the     discount. 

prospects  of  the  company. 

If  the    business   is   success- 
ful, the  profits  (dividends)  are  The  profits  of  the  company  at  the 

T    •  1   J  J-        i.     xi  end  of  the  first  year  were  ft 660,000, 

divided  according  to  the  num-  ,     ,.  .,     ,    ,«„/         I  \     a 

°  and  a  dividend  of  6  %  was  declared. 

ber  of  shares. 

Brokers  buy  and    sell   stock  After  the  dividend,  James  Lyman 

for   their   employers,  charging  bought  the  50  shares  of  John  Fluker, 

something  (brokerage)  both  for  through  a  broker,  at  $110  a  share, 

buying  and  selling.  P^^'^g  *  %  ^'^^^^^^^e. 

1.  What  is  the  number  of  this  certificate  of  stock  ? 

2.  To  whom  was  the  certificate  issued  ?   For  how  many  shares  ? 
a   ^Vhat  was  the  par  value  of  each  share?     How  do  you 

know  ? 

4.  What  was  the  market  value  of  each  share  at  the  time 
of  issue  ?     How  do  you  know  ? 

5.  If  the  earnings  were  $  660,000,  what  dividend  might  have 
been  paid  on  each  share  ? 

6.  How  many  dollars  were  paid  as  dividend  on  each  share  ? 

7.  What  dividend,  in  all,  did  John  Fluker  receive? 

a   What  was  the  par  value  of  each  share  after  the  dividend  ? 
Does  the  par  value  ever  change  ? 

9.   What  was  the  market  value  of  each  share  after  the  divi- 
dend ?     Does  the  market  value  change  ? 

la  What  was  John  Fluker's  entire  gain  from  the  stock  ? 

11.  What  %  brokerage  did  James  Lyman  pay?  What  part 
of  a  dollar  on  each  share,  if  the  brokerage  is  some  %  of  the  par 
value? 

12.  How  much  brokerage  in  all  did  James  Lyman  pay  ?  What 
was  the  entire  cost  of  the  stock  to  James  Lyman  ? 


208  PROBLKMS 

PROBLEMS 

The  par  value  of  one  share  is  f  lOOy  unless  otherwise  stated. 

This  understanding  saves  confusion,  and  makes  it  unnecessary  to  call 
attention  to  the  par  value. 

The  cost,  the  selling  price,  the  dividend,  the  brokerage,  each  is 

some  %  of  the  imr  value. 

These  tenns  must  be  some-  per  cent  either  of  the  market  value  or  of  the 
par  value.  The  par  value  never  changes,  the  market  value  is  constantly 
changing ;  hence  the  former  is  selected. 

All  examples  in  stocks  may  be  analyzed  by  using  $1, 1%,  or  1 
share  as  the  unit.  The  last  is  the  simplest,  because  it  eliminates 
%  as  much  as  possible. 

The  student  should  be  able  to  declare  in  dollars  the  cost  of  one 
share,  the  selling  price  of  otie  sliare,  the  dividend  on  one  sliare,  and 
the  brokerage  of  one  share,  however  the  same  is  stated. 

la   Cost.     What  is  the  cost  of  1  share  of  6%  stock  at  60  ? 

Ans.  9  60.    The  cost  is  60  %  of  $  100,  or  $  60. 

14.  Selling  price.  What  is  the  selling  price  of  1  share  of  8% 
stock  at  70? 

Ans.  $  70.    The  selling  price  is  70  %  of  $  100,  or  $  70. 

15.  Dividend.     What  is  the  dividend  on  1  share  of  6%  stock  ? 
Ans.  $  6.     The  dividend  is  6  %  of  $  100,  or  $  6. 

16.  Brokerage.  What  is  the  brokerage  on  1  share  of  6%  stock 
ati%? 

Ans.  $  |.    The  brokerage  is  |  %  of  $  100,  or  $  |. 

Note. — On  the  New  York  Stock  Exchange,  brokerage  higher  than  4%  is  not 
allowed. 

How  many  shares  in  : 

17.  $600    6%  stock?  19.  $1200    2%  stock? 

la   $800    3%  stock?  20.   $1500    8%  stock? 

Ex.  17.  6  shares.  The  par  value  of  one  share  is  $  100  ;  ^600  is  the  par 
value  of  as  many  shares  as  $  100  is  contained  times  in  $600,  or  6  shares. 


STOCKS   AND  BONDS  209 

Brokerage  \ffoyfind  the  cost  of: 

21,  $4()0stock  at75.  2a  $  800  stock  at  80. 

22.  $320  stock  at  60.  24.   $040  stock  at  90. 

Ex.  21.  1300.50.  The  entire  cost  of  1  share  is  $  75^  ;  the  cost  of  4  shares 
is  4  times  975|,  or  $300.50. 

Brokerage  \%,find  the  net  proceeds  of: 

25.  $400  stock  sold  at  lb\.  27.  $  800  stock  sold  at  80. 

26w  $  600  stock  sold  at  90^.  2a  $  240  stock  sold  at  60. 

Ex.  25.  $300.  The  net  proceeds  of  1  share  is  $75^  -  $  J,  or  $75  ;  of  4 
shares,  4  times  §75,  or  $300. 

Find  the  dividend  on : 

29.  $  400  3% 's  bought  at  80.  31.  $  1200  4%'s  sold  at  75. 

30.  $  300  6% 's  bought  at  00.  32.  $  1500  5%'s  sold  at  90. 
Ex.  29.  $  12.  The  dividend  on  1  share  is  $3;  on  4  shares,  4  times  $3,  or  $  12. 

How  many  shares  may  be  bought  for : 

3a  $900  at  89i,  brok.  \%?  35.  $1200  at  59J,  brok.  J%? 

34.  $600  at  74f,  brok.  \%7  36.  $1600  at  79|,  brok.  \%? 

Ex.  33.  10  shares.  The  cost  of  1  share  is  $90;  $900  will  buy  as  many 
shares  as  $90  is  contained  times  in  $900,  or  10  shares. 

How  much  stock  gives  an  income  of: 

37.  $200;  stock  5%  ?  39.  $600;  stock  3%  ? 

3a  $400;  stock  4%  ?  40.   $800;  stock  8%  ? 

Ex.  37.  $4000.  The  income  on  1  share  is  $5  ;  ii  will  take  as  many  shares 
to  yield  $200  as  $5  is  contained  times  in  $200,  or  40  shares.  40  shares  = 
$4000  stock. 

What  %  mil  I  realize  on  my  investment  when : 

41.  5%'s  are  bought  at  40  ?  43.  3% 's  are  bought  at  60  ? 

42.  8%'8  are  bought  at  60  ?  44.  4%'s  are  bought  at  80? 

Ex.  41.  12^  %.  One  share  costs  $40  and  gains  $5 ;  the  gain  %  is  $5+$40, 
orl2J%. 

▲MKR.    ARITH.  — 14 


210  PROBLEMS 

45.   Find  the  cost  of  $6000  S%  stock  at  80,  brokerage  J%. 

Given  terms:  no.  shares,  60 ;  first  cost  of  1  share,  $80;  brokerage  on  1 
share,  $  J. 

§80  =  cost  2  sh. 

-  =  brok.  I  sh.  Since  the  first  cost  of  1  share  Is 
8                                              $80,  and  the  brokerage  $  ^,  the  en- 

I  _       .  tire  cost  is  the  sum,  or  $80 J.    The 

80^  -  entire  c.  1  sh.  ^^^  ^^  ^  ^^^^^  ^  ^  ^^^  ^  ^^^  ^^ 

^  $4006.25. 


$4006.i5  =  entire  c.  all. 

46.   Find  the  proceeds  of  $5000  3%  stock  at  80,  brokerage  J%. 
Given  terms :  first  sell.  p.  1  share,  $  80  ;  brokerage  1  share,  $  ^. 
§80  =  sell.  p.  1  sh. 

i  _  brok.  1  sh.  Since  the  selling  price  of  1  share 

8  is  $80,  and  the  brokerage  $|,  the 

^  proceeds  will  be  the  difference,  or 

79-^=  proceeds.  ^^g^       ^Pl^^  proceeds  of  60  shares 

60  will  be  60  times  $ 79|,  or  $  3993.76. 


$3995.75  =  entire  proc. 

47.   How  much  3%  stock  (50)  can  be  bought  at  98  for  $  2456.25, 
brokerage  J%  ? 

Given  terms :  first  cost  1  share  $  49  ;  brokerage  on  1  share,  $  |. 

f  49.       =  cost  1  share. 

^125  —  brokerage.  Since  the  first  cost  of  1  share  is 

$49,  and  the  brokerage  $  J,  the  en- 


$ 49.185  =  entire  cost  1  share.  ;.     '       .    .„  .,„  ^ ^^«^  oW      a<. 

49.2e5)24f6.S5{50  ^        ^^^^^  ^^^  ^  ^^^^j^^^  ^^ 

z4o6  z5 

4a  What  is  the  dividend  on  $  2500  3%  stock  (50)  ? 

Given  terms:  dividend  on  1  share,  $  1.60  ;  no.  of  shares,  60. 

ft  1.50  Since  the  dividend  on  1  share  is 

50  ■    $1.60,  on  60  shares  it  is  60  times 

1 75  $1.60,  or  $76. 


STOCKS   AND   BONDS  211 

49.  How  much  5%  stock  at  80,  must  be  bought  to  get  an  annual 
income  of  $240? 

Given  «cn»« .•  dividend  on   1   share,  $5;  entire  income,  $240;  cost  of 

1  share,  9  80. 

Since  the  gain  on  1  share  is  $5,  it  will  take 
5)g40  as  many  shares  to  gain  $240,  as  $5  is  con- 

^  tained  times  in  $  240,  or  48  shares,  or  $  4800 

stock. 

50.  Find  the  rate  per  cent  of  dividend  when  $  6200  stock  yields 
$310. 

Given  terms  :  no.  shares,  62  ;  entire  income,  $  310. 

Since  02  shares  yield  an  income  of  $310, 
6SM10{5  1  share  will  yield  gJj  of  $  310,  or  $  5. 

SIO  Since  the  dividend  on  1  share  is  $5,  the 

stock  is  6%  stock. 

51.  If  3%  stock  is  at  80,  what  rate  %  will  a  person  receive  on 

his  investment  ? 

Given  terms:  investment  in  1  share,  $80 ;  dividend  on  1  share,  $3.     Re- 
lation :  income  =  B  x  investment. ' 

S  =  R  X  80 

g        ^  Since  the  income  is  some  %  of  the  invest- 

■R  =  —  =  ^'i  %  ment,  the  rate  is  $  3  -r-  $  80,  or  ^  %. 

52.  What  must  be  the  price  of  a  3%  stock  to  equal  a  4%  stock 
at  80? 

Given  terms :   income  on   1  share  tirst  stock,  $  3 ;    income  on  1  share 

second  stock,  $4;  investment  in  1  share  second  stock,  $80.  Relations: 
income  on  2d  =  ^  x  invest,  in  2d  ;  rate  on  first  investment  =  rate  on  second 
investment. 

4  =  Rx  80 

y,_  4  _  A(v  Invst.  =  —  zz60 

80      ^  .05 

5%  Invst  =  3  ^^**  ^  '^^^^  ^%  stock,  $60 

53.  What  must  be  the  price  of  stock  when  $8300  stock  is 
bought  for  $4150? 

Given  terms     no.  shares,  83;  cost  of  all,  $4150. 


212  PROHLEMS 

54.  What  is  the  cost  of  4000  3^%  government  bonds  ($  1000) 
at  01 1,  brokerage  J  %  ? 

55.  I  buy  200  shares  railroad  stock  at  102J,  and  sell  them  at 
104 J,  brokerage  J%  in  each  transaction.     How  much  do  I  gain? 

56.  My  broker  buys  for  me  300  New  York  state  bonds  at  72^, 
brokerage  J%.     How  much  do  the  bonds  cost  me? 

57.  What  are  the  proceeds  from  the  sale  of  45  shares  of  Ohio 
state  bonds  at  93,  brokerage  |%. 

5a  Find  the  proceeds  of  $  1000  New  York  city  4J's  at  101  J, 
brokerage  J%. 

Note.  —  N.  Y.  city  41*8  means  New  York  city  bonds  yielding  4i%  dividend. 

59.  What  are  the  proceeds  from  the  sale  of  80  shares  of  Ohio 
state  bonds  at  90,  brokerage  J  %  ? 

6a  If  Altar  mining  stock  sells  for  83J,  what  are  the  proceeds 
from  150  shares,  brokerage  i%  ? 

61.  How  many  shares  of  stock  at  62 J,  brokerage  J%,  can  be 
bought  for  $  9337.50  ? 

62.  A  broker  invested  $24,062.50  in  Union  Pacific  bonds  at 
4%  discount,  brokerage  J%.  How  many  shares  of  stock  will  his 
principal  receive  ? 

63.  A  pays  a  debt  of  $  2045  with  bank  stock  at  102^.  How 
much  stock  w^ill  the  creditor  receive  ? 

64.  The  Altar  Mining  Company  declares  a  dividend  of  6J%. 
How  much  will  a  stockholder  receive  who  holds  50  shares  ? 

65.  What  income  will  be  realized  from  investing  $58,080  in 
4J%  stock  at  90 J,  allowing  \%  for  brokerage? 

66.  A  speculator  invested  $2400  in  mining  stock  which  pays 
him  a  dividend  of  4J%.  If  he  agrees  to  take  his  dividend  in  coal 
at  6^  a  bu.,  how  many  bushels  should  he  receive  ? 

67.  Stock  bought  at  93 J^  and  yielding  5%,  bears  $225  annual 
income.     How  many  shares  are  there? 

6a  A  man  receives  $882  as  a  7%  dividend  on  his  stock. 
How  many  shares  does  he  hold  ? 


STOCKS  AND  BONDS  213 

69.  How  much  must  be  invested  in  U.  S.  4's  at  121  J,  brokerage 
J%,  that  I  may  pay  a  debt  of  $900  from  one  year's  income  ?  ' 

70.  If  I  liave  to  pay  an  assessment  of  $G10  on  305  shares  of 
stock  (50),  what  is  the  rate  of  assessment  ? 

71.  I  paid  $  1G,G05  for  stock  at  92,  brokerage  \%.  If  it  yields 
$3000  the  first  year,  what  is  the  rate  of  dividend  ? 

72.  The  earnings  of  a  mining  company,  whose  capital  is 
$120,000,  amount  in  one  year  to  $24,000;  their  expenses  are 
$  9000.     What  rate  of  dividend  can  they  declare  ? 

73.  What  per  cent  on  my  investment  will  I  realize  by  buying 
6%  stock  at  115? 

74.  If  1  invest  $  18,000  in  U.  S.  4's  at  102,  my  annual  income 
is  what  per  cent  of  my  investment  ? 

75.  Philadelphia  G's  are  bought  at  81}.  What  is  the  rate  of 
investment? 

76.  How  much  must  a  broker  pay  for  Virginia  G's  to  realize 
S%  on  his  investment  ? 

77.  What  must  be  the  price  of  3J%  stock  to  equal  4}%  stock 
at  90? 

7a  What  must  be  the  market  value  of  60  shares  4%  stock  to 
equal  G%  stock  at  75? 

79.  A  man's  rate  of  dividend  is  4%.  What  must  be  the  price 
of  stock  if  he  realizes  3%  on  his  investment? 

80.  What  income  is  derived  from  $500  G%  stock  (25)  ? 

81.  A  sells  GO  shares  of  stock  at  80,  and  invests  the  proceeds 
in  sto(^k  at  75.  If  the  latter  yields  4%  annually,  what  is  his 
income? 

82.  I  own  500  shares  mining  stock  bought  at  102.  If  the  stock 
yields  3%  annually,  what  is  my  income  in  dollars?  What  |)er 
cent  of  my  investment  is  my  income  ? 

83.  What  is  the  better  investment  for  $  1500,  6%  stock  at  75, 
or  4%  stock  at  60?  What  is  the  difference,  in  dollars,  in  the 
annual  income? 


214  PROBLEMS 

84.  A  stockholder  owns  3%  railroad  stock  worth  90,  and  2J% 
mining  stock  worth  70.  If  his  income  from  each  is  $600  a  year, 
in  which  stock  has  he  the  larger  investment  ? 

85.  How  much  stock,  brokerage  J%,  must  be  sold  at  137 J  to 
buy  with  the  proceeds  $3600  of  stock  at  68 J,  brokerage  \%? 

86.  A  man  receives  an  annual  income  of  $270  from  4J%  stock, 
and  $  160  from  2%  stock.  If  he  sells  the  former  at  85,  and  half 
the  latter  at  95,  brokerage  i%  in  each  transaction,  how  much 
does  he  receive  ? 

87.  If  4%  stock  is  at  60,  what  rate  per  cent  will  a  person  re- 
ceive on  his  investment  ? 

8a  A  man  invested  a  certain  sum  of  money  in  5%  stock  at  80, 
and  twice  as  much  in  4%  stock.  If  his  income  from  the  former 
is  $  300,  and  from  the  latter  1 J  times  as  much,  what  was  the  price 
of  a  share  in  the  latter  investment  ? 

89.  B's  income  on  80  shares  of  stock  is  $  320.  If  this  yields 
him  5%  on  his  investment,  what  is  his  entire  stock  worth? 

90.  A  man  owns  70  shares  2J%  stock  and  45  shares  4%  stock. 
Which  is  worth  more  in  market,  and  by  how  much,  if  the  rate 
realized  on  each  investment  is  5%  ? 

91.  \Yhat  amount  must  be  invested  in  3%  stock  at  90  to  yield 
an  annual  income  of  $  225  ? 

92.  By  selling  60  shares  of  4%  stock  at  90,  and  investing  the 
proceeds  in  other  stock  at  75,  my  income  is  increased  by  $  84. 
What  is  the  entire  income  from  the  latter  investment  ? 

93.  A  man  owns  $8000  5%  insurance  stock.  He  exchanges 
this  for  mining  stock  at  80,  which  increases  his  income  $150. 
What  rate  per  cent  dividend  does  the  mining  stock  yield  ? 

94.  My  income  on  70  shares  of  stock  is  $  245.  What  rate  of 
dividend  does  the  stock  pay  ? 

95.  I  sell  $9000  5%  stock  at  80,  and  invest  the  proceeds  in 
S%  stock  at  60.  I  increase  or  decrease  my  income,  and  by  how 
much? 


STOCKS   AND   BONDS  216 

DIFFICULT   PROBLEMS 

In  some  problems,  it  is  best  to  represent  a  required  term  by  x, 

96.  Suppose  10%  state  stock  is  20%  better  in  market  than 
4%  railroad  stock.  If  A's  income  is  $500  from  each,  how  much 
money  has  he  paid   for  each,  the  whole  investment  bringing 

Given  terms:  income  on  1  share  state  stock,  $10;  income  on  1  share  rail- 
road stock,  $4  ;  whole  income  on  state  stock,  $500 ;  whole  income  on  K.  U. 
stock,  $600. 

lielations:  cost  1  share  state  stock  =  120%  cost  1  share  R.  R.  stock. 
G,§j  %  of  investment  =  $  1000. 

Let  X  =  cost  1  share  R.  R.  stock,  $  90 

125  X  =  entire  cost  R.  R.  stock,  $  11250 

1.2  X  =  cost  1  share  state  stock,  $  108 

60  X  =  entire  ajst  state  stock,  $  5400 

185  X  =  entire  investment 

6A%of  185x  =  1000 

a:  =  90 

97.  I  received  a  10%  stock  dividend,  and  then  had  102  shares 
(1^50  each)  and  $15  of  another  share.  How  many  shares  had  I 
before  the  dividend  ? 

9a  Thomas  Reed  bought  6%  mining  stock  at  114|,  and  4% 
furnace  stock  at  112 J,  brokerage  J%.  The  latter  cost  him  $430 
more  than  the  former,  but  yielded  the  same  income.  How  much 
did  each  cost  him  ? 

99.  I  bought  stock  at  10%  discount,  which  rose  to  5%  premium, 
and  sold  for  cash.  Paying  a  debt  of  $33,  I  invested  the  balance 
in  stock  at  2%  premium,  which,  at  par,  left  me  $11  less  than  at 
first.     How  much  money  had  I  at  first  ? 

100.  W.  T.  Baird  invested  a  certain  sum  of  money  in  Phila- 
delphia G's  at  115J,  and  three  times  as  much  in  Union  Pacific 
7's  at  89 J,  brokeraj^e  ]^%  in  each  case.  How  much  was  invested 
in  each  kind  of  stock,  if  the  annual  income  was  $  9920  in  all  ? 


INTEREST 


TERMS  AND  RELATIONS 


Money  paid  for  the  use  of 
money  is  interest;  the  money 
loaned  is  the /)r//<c(/>a/;  the  sum 
of  the  principal  and  the  interest 
is  the  amount. 

1.  If  $  100  is  loaned  for  3  yr. 
at  6%,  and  no  payment  is  made 
until  the  end  of  the  third  year, 
how  much  interest  is  then  due  ? 

There  are  three  conceptions  : 

That  the  principal  alone  bears 
interest,  simple  interest.  * 

That  the  principal,  and  the 
interest  on  the  principal  at  the 
end  of  each  year,  bear  interest, 
annual  interest. 

That  the  principal,  the  inter- 
est on  the  principal  at  the  end 
of  each  year,  and  all  other  in- 
terest at  the  end  of  each  year, 
bear  interest,  compound  interest. 

Unless  otherwise  stated, 
simple  interest  is  always  un- 
derstood. 


Illi:stration 
If  $6  is  paid  fur  the  use  of  9  100, 
$  100  is  the  principal;  $0,  the  inter- 
est; 9  100,  the  amount. 

Simple  Interest 
1  yr.         2yr.         3  yr.         Total. 
f6  f6  f6  $18.00 

The  interest  on  $100  is  $0  each 
year. 

Annual  Interest 
1  yr.       2  yr.  3  yr.  Total. 

$6         f6  f6 

.36             .36 
_         .36         

f6  $6.36  $6.72  $19.08 
The  interest  on  $100  is  $6  each 
year.  The  interest  on  the  first  $  6  is 
36^  the  2d  year,  and  30  f  the  3d  year. 
The  interest  on  the  second  §6  is  36^ 
the  3d  year. 

Compound  Interest 
1  yr.     2  yr.  3  yr.  Total. 

$6      $6  $6 

.36  .36 

.36 
.0216 


$6      $6.36      $6.7416      $19.1016 
In  addition  to  the  annual  interest, 
the  first  36^  gains  2. 16  ^ 


216 


INTEREST 

SIMPLE  INTEREST 
First  Conception  :  The  principal  cdoue  bears  interest, 
2.  What  is  the  interest  of  $1 


217 


for  1  yr.  at  G%  ? 

3.  What  is  the  interest  of  $1 
for  1  mo.  at  6%? 

4.  What  is  the  interest  of  $  1 
for  1  da.  at  6^  ? 


The  interest  of  $  1  for  1  yr.  at 
6%  is  .06  of  a  1,  or  0^ 

Since  the  interest  of  $1  for  12 
mo.  isO)?,  for  1  ino.,  it  is  j^i  of  0^, 
or  o  m. 

Since  the  interest  of  f  1  for  30 
da.  is  5  m.,  for  1  da.,  it  is  ^^  of 
5  m.,  or  ^  of  a  mill. 


To  BE  memorized:   The  interest  of  $  1  for  1  year  at  6fo  *'*  ^^; 
for  1  monthy  ^ofa  cent;  for  1  day,  \of  a  mill. 

5.  What  is  the  interest  of  $  1  for  5  yr.  5  mo.  16  da.  2kt6%? 

f.30 

.025  The  interest  of  §  1  for  5  yr.  at  6  %  is  .30 ;  for 

QQ2*  5  mo.,  $.025;    for  10  days,  $.002|;  for  the 

^  whole  time,  f  .327f. 

.327\ 

6.  What  is  the  amount  of  $  3G0  for  5  yr.  5  mo.  16  da.  at  6%  ? 
^360 


.327\ 


117.96,    interest 
360. 


Assume  $1.  The  interest  of  $1  for  6  yr 
6  mo.  10  da.  at  6%  is  $.327| ;  for  $300,  300 
times  «.327i{,  or  $117.96.  The  amount  is 
$477.96. 


477.96,    amount 

Find  the  interest  of  f  1  at  6% 

7.  For  4  yr.  2  mo.  16  da. 

8.  For  5  yr.  7  mo.  9  da. 

9.  For  3  yr.  1  mo.  17  da. 

10.  For  7  yr.  7  mo.  7  da. 

11.  For  2  yr.  6  mo.  12  da. 


Find  the  amt.  of  f  125  at  6%  : 

12.  For  2  yr.  2  mo.  2  da. 

13.  For  6  yr.  6  mo.  6  da. 

14.  For  5  yr.  0  mo.  19  da. 

15.  For  5  yr.  7  mo.  27  da. 

16.  For  4  yr.  1  mo.  20  da. 


218  SIMPLE  INTEREST 

If  the  rate  is  not  6%,  the  interest  may  be  found  at  6  per  cent, 
and  the  result  modified  for  the  required  rate. 

17.   What  is  the  amount  of  $  3G0  for  3  yr.  3  mo.  5  da.  at  5% ? 
tS60 
'195'1  Assume  $  1.    The  interest  of  $  1 

for  3  yr.  @  6%  is  $.18 ;  for  3  mo., 


70.oO  @  6%.  ^.015;    for  6  da.,  $.000^  ;   for  the 

ii.75  @  1%.  whole  time,  $.196J. 

fro'rr  f7i,Krrt  'I^e  interest  of  f  360  is  360  times 

58,75  @  5%.  ^^^^^^  ^^  ^^^^_  ^^  gy^  ^  ,^^^  ^^ 

^^0.  $68.76. 


fjfl8.75y  amount, 

la  How  would  you  modify  the  interest  @  6%  to  find  the 
interest  at  8%?  at  10%?  at  7%?  at  9%? 

Am.  At  8  %,  I  would  add  a  third  of  the  interest,  because  8  is  6  plus  |  of 
6 ;  at  10  %,  I  would  divide  the  interest  by  6  and  multiply  by  10. 

When  the  time  does  not  exceed  4  months,  a  modification  of 
the  6%  method  is  in  general  use. 

Moving  the  decimal  point  2  places  to  the  left  in  the  principal,  gives 
the  interest  for  60  dat/s  at  6%. 

Proof  :  the  interest  of  $  1  for  60  da.  @  6%  is  1^,  or  yiff  o^  <>  1- 

19.  What  is  the  interest  of  $  167.80  for  3  mo.  3  da.  @  10%? 

91.678         60 

.889  80  The  interest  for  60  da.  @  6%  is 

.0839  S  %  1 .678  ;  for  30  da. ,  \  of  that,  or  $  .839  ; 

2.6009  @    6%  ^^^  ^  ^^^  t'o  of  that,  or  $.0839. 

4.334  @10% 

Find  the  interest  of  $438.96  for :  What  is  the  interest  of: 

20.  3  yr.  10  mo.  17  da.  @    6%.  25.  $  295.40  for  36  da.  @  5%  ? 

21.  4yr.    0  mo.  25  da.  @    9%.  26.  $360.20  for  93  da.  @  7%  ? 

22.  2  yr.    6  mo.  17  da.  @  10%.  27.  %  235.90  for  93  da.  @  9%  ? 

23.  6yr.    1  mo.  18  da.  @    7%.  28.  $ 840.50  for  33  da.  @  8 %  ? 

24.  4  yr.    3  mo.  12  da.  @  4J%.  29.  $ 501.02  for  63  da.  @  6%  ? 


INTEREST  219 

Accurate  interest,  that  is,  interest  found  by  counting  365,  in- 
stead of  360,  days  to  the  year  is  sometimes  computed. 

30.  Find  the  accurate  interest  of  $  625  for  80  days  at  6%. 

^^  X  .06  X  626  =  8.219.  Since  the  interest  for  366  days  is  .06 

S65      '  of  the  principal,  the  interest  for  80  days 

Accurate  Int.  =  ^8.219.         «  sVs  x  .06  x  $  625,  or  $  8.219. 

31.  Find  the  interest  of  $  625  for  80  days,  counting  360  days 
to  the  year,  subtract  ^  of  the  result,  and  observe  that  accurate 
interest  has  been  computed. 

32.  What  is  the  accurate  interest  of  $  100  from  Jan.  1,  1897, 
to  Jan.  1,  1898,  at  6%  ?  the  interest  as  commonly  found  ? 

Ana.  $6  in  each  case.  The  two  methods  agree  for  a  whole  number  of 
years  ;  for  a  fraction  of  a  year  accurate  interest  is  less. 

33.  Show  that  accurate  interest  for  any  number  of  days  les& 
than  a  year,  may  be  found  by  deducting  j^^  of  the  interest  found 
by  counting  360  days  to  a  year. 

Let  R  =  the  int.  for  1  yr. 

R 

=  int.  1  da.,  365  da.  to  ayr.  mt.  . .    ^i.    •  *      *  *      j  v 

^gj  '  ^  That  IS,  the  mterest  found  by 

n  counting  360  da.  to  the  year  mul- 

=  int.  1  da.f  860  da.  to  a  yr.        tiplied  by  \\,  or  diminished  by 

^^0  ^^  becomes  the  accurate  interest. 

^  x^   or-?-x^  =  -^ 
360     365'        360      73     365 

Find  the  accurate  interest  of : 

3C  $200  from  Dec.  13, 1896,  to  May  1, 1897,  @  6%. 

35.  1440  from  Jan.  20,  1896,  to  Apr.  5,  1897,  @  8%. 

36.  f  450  from  June  16,  1895,  to  Nov.  8,  18%,  @  10%. 

37.  $300  from  Dec.  1,  1806.  to  May  10,  1898,  @  4%. 


220  SIMPLE  INTEREST 

INDIRECT   CASES 
A»  a  basis,  always  assume  1  of  the  denomination  required. 
3a   Wliat  principal  will  gain  $  76.80  in  3  yr.  2  mo.  12  da.  @  8%? 

J8 

^01  Aasome  f  1.    $1  in  3  yr.  2  mo. 

S^n  )78  f^nOtSiOa         002        12  da.  @  8%  will  gain  $  .266  ;  it  will 

-^  iuit       as  $  .266  is  contained  times  in  9  76.80, 

'SM       or  $300. 

39.  At  what  %  will  $  300  gain  $  76.80  in  3  yr.  2  mo.  12  da.  ? 

,192  02  Assume  1%.    $  300  in  3  yr.  2  mo. 

S7.60     at  6%  'qoo  ^2  da.  at  1  %  will  gain  $  9.60  ;  it  will 

9.60     ati%  ^  ^^^^  ^  ""*"y  %  ^  ^'"  $76.80,  as 

J^^,-.^    ^x^  /O^  $9.60  is  contained  times  in  $76.80, 

9.60)76.80(8  -^^^  or  8%. 

76.80  *^ 

40.  In  what  time  will  $300  gain  $76.80  at  8%? 

300  3.2  yr. 

—                             ^^  Assume  1  yr.     $.300  in  1  yr.  @  8% 

^■*  2  J,  rnn  ^^^^  ^*"  $  24  ;  it  will  take  as  many 

24)76.80(3.2          "^       '  years  to  gain  $70.80,  as  $24  is  con- 

72                        80  tained  times  in  $  76.80,  or  3  yr.  2  mo. 

48                     12  da.  12  da. 

41.  What  principal  will  amount  to  $  376.80  in  3  yr.  2  mo.  12  da, 
at8%? 

.18 

Q2  Assume  $1.     $1  in  3  yr.  2  mo. 

002  12  da.  at  8%  will  amount  to  §  1.256  ; 

1.256)376.800(300       '—3  it  will  t.ake  as  many  dollars  to  amount 

3768                      'Vi')  to  $370.80,  as  $1,250    is  contained 

-0^^  times  in  $376.80,  or  $300. 
.256 


INTEREST  221 

Each  example  shouhl  be  changed  to  a  form  already  given. 

42.  At  what  %  will  $300  amount  to  $37G.80  in  3  yr.  2  mo. 
12  da.  ? 

This  means,  "  At  what  %  will  $  300  gain  $  76.80  in  3  yr.  2  rao,  12  da.  ?  " 
See  Ex.  39. 

4a  In  what  time  will  $  300  amount  to  $  376.80  at  8%  ? 

This  means,  "  In  what  time  will  ^  300  gain  $ 76.80  at  8  %  ?  "     /S^ee  i^x.  ^0. 

44.  At  what  %  will  a  sum  triple  in  30  years  ? 
This  means,  "  At  what  %  will  $  10  gain  $ 20  in  30  yr.?  " 

45.  In  what  time  will  a  sum  quadruple  at  6%? 
This  means,  "  In  what  time  will  $  10  gain  ^  30  @  6  %  ?  " 

46.  Is  it  proper  to  reason  thus :    "  Since  $  1  amounts  to  $  1.06, 
$  5  will  amount  to  5  times  $  1.06,  or  $  5.30  ? 

Yes.     Because  $5  will  amount  to  5  times  as  much  as  $  I. 

47.  Is  it  proper  to  reason  thus:    "If  a  principal  amounts  to 
9  1.06  in  1  yr.,  in  2  yr.  it  will  amount  to  2  times  $  1.06,  or  f  2.12  ?  " 

No.    The  amount  is  always  once  the  principal  plus  the  interest. 

4a  At  what  %  will  a  sum  double  in  10  years  ? 

49.  In  what  time  will  $  260  gain  $  29.90  at  5%? 

5a  In  what  time  will  a  sum  triple  at  10  per  cent  ? 

51.  In  what  time  will  $  260  amount  to  $  289.90  at  5%? 

52.  At  what  %  will  $  260  gain  $  29.90  in  2  yr.  3  mo.  18  da.  ? 

53.  At  what  %  will  $  260  amount  to  $  289.90  in  2  yr.  3  mo.  18  da.  ? 

54.  Whatprincipal  willgain$29.90in2yr.3mo.  18da.@5%? 

55.  What  principal  will  amount  to  $  289.90  in  2  yr.  3  mo.  18  da. 


222  SIMPLE   INTEREST 

MISCELLANEOUS 
Find: 

56.  Interest  of  $  1025  for  3  mo.  6  da.  @  9%. 

57.  Amount  of  $  2400  for  6  yr.  5  mo.  20  da.  @  6%. 
5a  Amount  of  $  930  for  10  yr.  3  mo.  24  da.  @  4%. 
59.  Ainouut  of  $  150  from  Apr.  4  to  Dec.  18  @  8%. 

What  principal : 

GO.  Will  produce  $  68.20  interest  in  2  yr.  7  mo.  @  4%? 

61.  Will  produce  $  1610  interest  in  4  yr.  5  mo.  20  da.  @  6%? 

62.  Will  amount  to  $  742  in  6  yr.  18  da.  @S%? 

63.  Will  amount  to  $  1065.75  in  10  mo.  15  da.  @  10«i^.? 


At  what  rate  per  cent : 

64.  Will  $  824  amount  to  $  957.90  in  3  yr.  3  mo.  ? 

65.  Will  $  235  produce  %  84.60  interest  in  6  yr.  ? 

66.  Will  $900  amount  to  $  1200  in  10  years? 

67.  Will  $  840  produce  $  70.84  interest  in  4  yr.  2  mo.  18  da.  ? 

Find  the  time  in  which  : 

6a  $  286  Avill  gain  $  70.07  @7%. 

69.  $  800  will  amount  to  $  1040  @  6%. 

70.  $  340  will  produce  $  13.60  interest  @  5%. 

71.  $760  will  gain  $285  interest®  10%. 

What  will  be  the  amount  of: 

72.  $  100  in  3  yr.,  if  the  amount  in  1  yr.  is  $  125  ? 

73.  $  80  in  5  yr.,  if  the  amount  in  2  y r.  is  $  100  ? 

74.  $  160  in  6  yr.,  if  the  amount  in  2  yr.  is  $  192  ? 

75.  $  450  in  10  yr.,  if  the  amount  in  3  yr.  is  $  595  ? 


INTEREST 


223 


PARTIAL  PAYMENTS  — LONG  NOTES 


A  written  promise  to  pay  a  sum  of 
money  on  demand  or  at  a  stated  time, 
is  a  note ;  the  sum  promised,  the  jnin- 
cipal,  or  face  of  the  note ;  the  time 
when  a  note  is  due,  its  mcUurity ;  the 
person  to  whom  the  money  is  to  be 
paid,  the  payee ;  the  person  who  signs 
the  note,  the  maker  or  drawer. 

Whenever  a  payment,  called  an 
indorsement,  is  made,  the  date  and 
amount  of  the  payment  are  written 
upon  the  back  of  the  note. 


June  4,  1893,  Harold  Blake 
bought  a  farm  of  Peter  Ford 
for  $1000,  paying  $475  cash 
and  giving  Note  No.  1  for  the 
balance.  Mr.  Blake  agrees  to 
pay  the  principal,  with  interest 
at  6%  from  date,  on  June  4, 
1890.  He  may  delay  payment 
as  long  as  Mr.  Ford  will  con- 
sent. 

Thedratccr  is  Harold  Blake ; 
the  payee^  Peter  Ford.  Mr. 
Blake  paid  $114.20,  Sep.  9, 
1894;  $8.29,  May  16,  1896; 
$244.38,  Aug.  6,  1896. 


f  625.00.  Emporia,  Kans.,  fun&  f,  189  3. 

<5kv&&  if&cLva^  after  date  J  promise  to  pay  to 

the  order  of ^eC&v  c^<>^<^-.,^...^,ww^.wv^.^^.c^^Vi'=' 

AuTtdvtd  tn^&ntif-fCve,.,..^^.,^....^^^Dollars,  with  interest 

a^  6%,  at  ^/i&  ^ivot  o^atuynat  Bank,  of  (^At^ttO'.^.s.^^ 

Value  received. 

Ko.  1.      Due  fun&  ^,  /S^6. 


^i  ^  ^i 

? :;  ;| 

S  =0  S 

On  ».»  "O 


d.  S'  ^ 

Co    >J    '^ 


224  SIMPLK   INTEREST 

The  United  States  rule  for  partial  payments  is  derived  from 
the  decision  of  the  Supreme  Court,  for  finding  the  amount  due  on 
a  note  when  payments  have  been  made. 

Unitkd  States  Rule 

Find  the  amount  of  the  principal  to  the  time  of  the  first  paj/mcnt; 
if  the  payment  equals  or  exceeds  the  interest^  subtract  the  payment 
from  the  amount  a}id  treat  the  remainder  as  a  new  princijxtl. 

If  the  jxiyment  is  less  than  the  interest,  find  the  amount  of  the 
same  princijxtl  to  the  time  when  the  sum  of  the  payments  shall  equal 
or  exceed  the  interest  due,  and  subtract  the  sum  of  the  payments  from 
the  amount. 

Proceed  in  the  same  manner  with  the  remaining  payments  until 
the  time  of  settlement, 

76.   What  was  due  on  note  No.  1,  Feb.  9, 1898  ? 

$525.00,  P. 

S9.8U  L  »/p,  ^94 
564^81 
114.20 
450.61,  P. 

18.48*  I.  5/15,  '95 
469.09 
SS.12,  I.  s/c,  '96 
NoTK.  —  By  arranging  the  dates  as  above,  errors  503. 21 

in  subtracting  will  be  discovered.     Each  date  is  252.67,  Sum  pay. 

subtracted  from  the  next  below.    The  difiference  249.54    P. 

between  the  first  and  last  dates  shoald  equal  the  Sg  58 

$272.12,  Due  2/^?,  '98 


189S  6    4 

1894  9    9 

1  S    5 

$114,20 

1895  5  15 

8    6 

8,29 

1896  8    6 

1  2  21 

244-98 

1898  2    9 

16    3 

4  8    5        4  8    6 


sura  of  the  differences. 


*  Since  the  interest  exceeds  the  payment,  we  find  the  interest  on  $  450.61  and  not 
on  $4G9.09,  for  1  yr.  2  nao.  21  da. 

77.   On  note  No.  2,  how  much  remained  due  Sept.  25, 1897  ? 

7a   On   note  No.  3,  liow  much   remained   due   Oct.  9,  1897, 
counting  interest  at  6%? 


INTEREST  226 

$750.00.  Boston,  Mass.,  June  17.  1892. 

On  demand,  for  value  received,  I  promise  to  pay  John  E.  Wiley, 
or  order,  seven  hundred  fifty  ^Y(r  dollars,  with  interest  at  6%. 
No.  2.  J.  J.  Hill. 

Payments:  Mar.  1,  1893,  075.50;  June  11,  1803,  $165;  Sept.  15,  1803, 
$101;  Jan.  21,  1804,  047.25;  Mar.  12,  1895,  $12.50;  Dec.  6,  1805,  008; 
July  7,  1896,  0 169. 


$300.00.  Wilmington,  Del.,  Apr.  30,  1895. 

On  demand,  for  value  received,  I  promise  to  pay  G.  R.  Bell,  or 
order,  three  hundred  -f^j^  dollars,  with  interest. 

No.  3.  Russell  Hibbs. 

Payments:  June  27,  1896,  0  150  ;  Dec.  9,  1890,  0160. 


$  1500.00.  Cincinnati,  O.,  Jan.  10,  1895. 

On  demand,  for  value  received,  I  promise  to  pay  Ernest  Buck- 
man,  or  order,  one  thousand  five  hundred  dollars,  with  interest 

**^%-  '  L.H.  Taylor. 

No.  4. 

Indorsements:  Mar.  3,  1895,  0250;  May  20,  1895,  0300;  July  1,  1896, 
0125;  Nov.  15,  1895,  075. 


$625.00.  Empobia,  Kans.,  Dec.  15,  1895. 

On  demand,  for  value  received,  I  promise  to  pay  James  Tyner, 
or  order,  five  hundred  twenty-five  dollars,  with  interest  at  8%. 
No.  6.  W.  N.  Simpson. 

IndorsemenU:  Jan.  16,  1896,  076  ;  Apr.  20.  1896,  060;  June  16,  1896, 
0140.60;  Aug.  1,  1896,  080;  Sept.  25,  1896,  0120. 

Notk.  —  For  treatment  of  No.  4  and  No.  5,  see  next  page. 

AMBB.    ABITH.  — 15 


226  SIMPLE  INTEREST 

PARTIAL  PAYMENTS  -  SHORT   NOTES 

When  notes,  upon  which  payments  have  been  made,  are  settled 
within  a  year  of  date,  merchants  commonly  disregard  the  U.  S. 
rule. 

Mbrcantile  Rulb 

'  Find  the  amount  of  eacJi  item  to  date  of  settlement. 

$  1728.00.  LiTTi.E  Rock,  Ark.,  Jan.  2,  1897. 

On  demand,  for  value  received,  I  promise  to  pay  William 
Chamberlain,  or  order,  one  thousand  seven  hundred  and  twenty- 
eight  ^5  dollars,  with  interest  at  0%. 

No.  6.  Henry  Kixg. 

Indorsements:  Mar.  1,  1897,  ^300;  May  16,  1897,  $150;  Sept.  1,  1897, 
#270;  Dec.  11,  1897,9135. 

79.  How  much  was  due  Dec.  13, 1897,  on  Note  No.  6  ? 

Principal, $1728.00 

Interest  to  Dec.  13,  1897,  345  da., 99.36 

$1827.36 

First  payment, $300.00 

Int.  to  Dec.  13,  1807,  287  da!,     .  14.35 

Second  payment, 150.00 

Int.  to  Dec.  13,  1897,  211  da.,     .  5.28 

Third  payment, 270.00 

Int.  to  Dec.  13,  1897,  103  da. ,    .  4.64 

Fourth  payment, 135.00 

Int.  to  Dec.  13,  1897,  2  da.,    .     .  .05 

879.32 

Bal.  due  Dec.  13,  1897, $948.04 

Owing  $1728  Jan.  2,  is  equivalent  to  owing  Dec.  13,  $  1728  +  the  interest 

of  $  1728  from  Jan.  2  to  Dec.  13,  or  $  1827.36. 

Paying  $300  Mar.  1,  is  equivalent  to  paying  Dec.  13,  $300  +  the  interest 

of  $300  from  Mar.  1  to  Dec.  13,  or  $314.35. 

eo.   On  Note  No.  4,  how  much  remained  due  Dec.  12,  1895  ? 
81.   On  Note  No.  5,  how  much  remained  due  Nov.  1,  1896  ? 


INTEREST  227 

TRUE   DISCOUNT 

The  true  present  woi-th  of  a  note,  or  of  a  sum  of  money  due  in 
the  future,  is  that  sum  which,  put  at  interest  now,  will  amount  to 
the  given  debt  at  the  expiration  of  the  time. 

82.  $  275  is  due  me  in  1  yr.  6  mo.  What  is  the  true  present 
worth,  interest  at  6%?     What  is  t]ie  true  discount  ? 

The  present  worth  is  that  sum  which,  put 

at  interest  to-day,  will  amount  to  $275  in 

1.09)275.00(252.29  \  yr.  6  mo.  at  6%. 

$275    face  $1  in  1  yr.  0  mo.  at  6%  will  amount  to 

^^^^^  ^1       $109;    it  will    take   as  many   dollars    to 

^52.29,  true  jyres.  worth     ^^^^^^  ^  ^  .^75  ^  3 1  Oj,  ia  contained  times 

$22.71,  discount  '^  *  275,  or  8  252.29. 

^  The  true  discount  is  $276  -  $252.29,  or 

$22.71. 
Note.— True  discount  is  rarely  found  on  notes  or  debts  running  less  than 
four  months. 

83.  Find  the  true  present  worth  and  true  discount  of  Note 
No.  7,  discounted  at  10%,  Mar.  1,  1896. 

^  300.00.  Emporia,  Kans.,  Jan.  2,  1896. 

Ninety  days  after  date,  value  received,  I  promise  to  pay  to  the 
order  of  William  Clarke,  at  the  First  National  Bank,  Three  hun- 
dred and  -^  Dollars,  with  interest  at  10%  after  maturity. 

No.  7.  W.  C.  Stevenson. 

Find  the : 

84.  True  present  worth  of  $  412  due  in  6  mo.,  int.  @  6%. 

85.  True  present  worth  of  $  324  due  in  8  mo.,  int.  @  12%. 

86.  True  present  worth  of  $  321  due  in  1  yr.  9  mo.,  int.  @  7%. 

Find  the  : 

87.  True  discount  of  $  590  due  in  2  yr.  8  mo.,  interest  at  9%. 

88.  True  discount  of  $  33G  due  in  3  yr.  10  mo.,  interest  at  12%. 

89.  True  discount  of  $  427  due  in  5  yr.  11  mo.,  interest  at  8%. 


228 


SIMPLE  INTEREST 


BANK  DISCOUNT 

In  discounting  notes,  bankers  use  the  following  rule : 

To  find  the  bank  discount,  compute  the  interest  on  the  amount  due 
at  maturity  for  (three  days  more  than*)  the  specified  time. 

To  find  the  proceeds,  subtract  the  bank  discount  from  the  amount 
due  at  maturity. 

90.   Find  the  bank  discount  at  10%  Jan.  2,  1896,  on  Note  7; 
iiud  the  bank  proceeds ;  find  the  date  of  maturity. 
fS.OO         60 

Since  the  note  does  not  bej^in  to  bear 
interest  until  U3  days  after  Jan.  2,  tlie 
amt.  due  at  maturity  is  the  face  of  the 
note,  or  $300. 

The  interest  of  $300  for  93  days  at 
10%  is  $7.75. 


1.60 
.15 


f4.65,    D.y      6% 

7.75,  D.y  10% 
f292,S5y  Proceeds 
Apr.  4)  <^«^€  ofmMurity 

91.  Write  Note  No.  7,  substituting  "with  interest  at  6%"  in 
place  of  "with  interest  at  10%  after  maturity."     Find  the  bank 
discount  and  proceeds  at  10%,  Feb.  2,  1896. 
fS.OO    60 


1.50 
.15 


Since  the  note  now  begins  to  bear 
interest  Jan.  2,  the  amount  at  maturity, 
or  the  face  value  of  the  note,  is  the 
amount  of  $300  for  93  days  @  6%,  or 
$304.65. 


f4.65y  Int. 
$804-65,  amt.  at  maiurity 

$3.0465     60 

.1015     f 

f  3.14s,  D.,    6% 

5.25,  D.,     10% 

$299.40,  Proceeds 

♦  Tills  phrase  should  be  stricken  out  for  the  following  states  and  territories  in 
which  the  three  days,  called  days  of  grace,  have  been  abolished  by  statute  :  Alas,, 
Cal..  Colo.,  Conn.,  D.  C.  Fla.,  Ida..  111.,  Me..  Md.,  Mass.,  Mont,,  N.  IL,  N.  J., 
N.  Y.,  N.  D.,  O.,  Ore.,  Pa.,  Utah,  Vt.,  and  Wis. 


Since  the  note  was  not  discounted 
until  Feb.  2,  or  31  days  after  date,  the 
specified  time  is  93  da.— 31  da.,  or  62  da. 


INTEREST  229 

92.  Write  a  bank  note  for  90  days,  with  interest  after  maturity, 
that  will  give  bank  proceeds  at  10  %,  of  $292.25. 

.01  60 

Assume  8  1.    Since  the  bank  proceeds 

of  «1  at  10% for  IW  da.  are  8. 074 J,  it  will  -000^        * 

take  as  many  dollars  to  yield  $292.26,  ni^t^       «% 

as  e  .974i  is  contained  times  in  8292.25,  '^  ^ 

or  8  300.  .025^      io% 

.974j)292M0{300 

93.  Find  the  date  of  maturity,  and  bank  discount  at  6  %,  Jan. 
10,  1896,  on  Note  No.  8. 

f  400.00  *  MiDDLETOWN,  CoNN.,  Jan.  10,  1896. 

Sixty  days  after  date,  value  received,  I  promise  to  pay  to  the 
order  of  the  Middlesex  County  National  Bank,  four  hundred  and 
^^  dollars,  with  interest  at  6  %  after  maturity. 

No.  8.  John  S.  Camp. 

94.  Find  the  bank  discount  at  6  %,  Mar.  3, 1896,  on  Note  No.  7. 

95.  Write  Note  No.  8,  substituting  "  with  interest  at  5  %  "  in 
place  of  "  with  interest  at  6  %  after  maturity,"  and  find  the  bank 
discount  at  6  %,  Feb.  19. 

96.  Find  the  true  discount  at  6  %,  of  Note  No.  8,  Jan.  10, 1896. 
How  much  does  the  banker  make  by  collecting  bank,  instead  of 
true  discount  ? 

97.  Show  that  the  difference  between  the  bank  discount  and  the 
true  discount  of  Note  No.  8,  Jan.  10, 1896,  is  the  interest  on  the 
true  discount  for  60  days. 

Note.  —  Do  not  count  3  da.  of  grace  in  either  case. 

9a  What  is  the  face  of  a  note,  which,  when  discounted  for 
2  mo.  24  da.  at  5  %,  will  yield  $224.00  bank  proceeds? 

99.  Write  a  bank  note  for  3  mo.,  maker,  John  Brown ;  payee, 
Henry  Short;  proceeds  at  6  %,  $  732.92. 


230 


SIMPLE  INTEREST 


SETTLEMENT  OF  ACCOUNTS  —  INTEREST  METHOD 

In  settling  accounts  between  wholesale  and  retail  dealers,  oi 
where  the  amounts  are  large  or  of  long  standing,  it  is  customary 
to  take  account  of  interest. 

100.  Thomas  Stone  bought  merchandise  of  us  as  shown  on  the 
Dr.  side  of  the  following  account,  and  made  payments  as  on  the 
Cr.  side.  Settlement  was  made  Sep.  15,  189G;  how  much  did 
Mr.  Stone  owe  at  that  time,  counting  interest  at  6%  ? 


Dr. 

Thomas  Stone 

Cr. 

189G 

1S96 

Mar.  7 

To  Mdse 

500 

Mar.J) 

By  Cash 

SOO 

Apr.    3 

To  Mdse 

SOD 

Apr.  1 

By  Cash 

200 

Aug.  2 

To  Mdse 

700 

July  I 

By  Cash 

500 

First  debt                  f 

500.00                    First  payment           $S00.00 

Int.  to  Sep.  16 

IG.OO                         Int.  to  Sep.  15 

9.50 

Second  debt 

SOO.                        Second  payment 

200.00 

Int.  to  Sep.  16 

8.g5                        Int.  to  Sep.  15 

5.57 

Third  debt 

7pO.                        Third  payment 

500.00 

lE 

t.  to  Sep.  16 

5.1 

S 

Int.  to  Sep.  15 

6.33 

§1529.38 
Bal.  due  Sep.  15,  1896,  $507.98. 


$1021.40 


Owing  $500  Mar.  7,  is  equivalent 
to  owing  Sep.  15,  8  500  + the  inter- 
est of  $500  from  Mar.  7  to  Sep.  15, 
or  $516.00. 

Owing  $300  Apr.  3,  is  equivalent 
to  owing  Sep.  15,  $300  + the  inter- 
est of  $300  from  Apr.  3  to  Sep.  15, 
or  $  308.25 ;  etc. 

The  pupil  should  finish  the  ex- 
planation. 


Paying  $300  Mar.  9,  is  equivalent 
to  paying  Sep.  15,  $300  +  the  inter- 
est of  $300  from  Mar.  9  to  Sep.  15, 
or  $309.50. 

Paying  $200  Apr.  1,  is  equivalent 
to  paying  Sep.  15,  $200  +  the  inter- 
est of  $  200  from  Apr.  1  to  Sep.  15, 
or  $  205.57  ;  etc. 

The  pupil  should  finish  the  ex- 
planation. 


Note.  —  This  method  is  identical  with  that  used  in  settling  short  time  notes, 
when  payments  are  made  within  a  year  of  date.    6'ee  p.  226. 


interp:st 


231 


EQUATED  TIME  METHOD 
103.   Solve  example  100  by  the  equated  time  method. 


Dr. 

500  X  192  =  96000 

SOO  X  165  =  49500 

700  y.    44=  S0800 


1600 


176S00 


1600 
1000 


600 


Cr. 

SOO  X  190  =    57000 

goo  X  167  =    SS400 

600  X    76=    SSOOO 

1000  128400 

176S00 
128400 
47900 


$500. 

7.98  Int.  f  47900  1  da. 


$607.98  Bal.  due  Sep.  16. 


Owing  $  600  Mar.  7,  is  equivalent 
to  owing  Sep.  16,  $  600  +  the  inter- 
est of  $600  for  192  days,  or  the 
interest  of  $  96,000  for  1  da. 

Owing  $.300  Apr.  3,  is  equivalent 
to  owing  Sep.  16,  $.300  +  the  inter- 
est of  $300  for  106  days,  or  the 
interest  of  $49,600  for  1  da. 


Paying  $  300  Mar.  9,  is  equivalent 
to  paying  Sep.  18,  $300  +  the  inter- 
est of  $.300  for  190  days,  or  the 
interest  of  $57,000  for  1  da. 

Paying  $200  Apr,  1,  is  equivalent 
to  paying  Sep.  16,  $200  +  the  inter- 
est of  $  200  for  167  days,  or  the  inter- 
est of  $  33,400  for  1  da. 


Sep.  16,  he  owes  $  1500  +  the  interest  of  $  176,300  for  1  da.,  and  has  paid 
$  1000  +  the  interest  of  $  128,400  for  1  da.  The  bal.  due  is  the  difference,  or 
$600  4-  the  interest  of  $47,900  for  1  da.,  or  $507.98. 


102.  Find  the  equated  time  for  the  payment  of  the  account  in 
example  100. 

By  the  equated  time^  is  meant  the  time  when  the  balance  of  the  account 
i$  due. 

Sep.  15,  Mr.  Stone  owed  $  500  -|-  the  interest  of  $  47,900  for  1  da.,  or  $  500 
+  the  interest  of  $500  for  as  many  days  as  $600  is  contained  times  in 
$47,1K)0,  or  for  96  da.  The  equated  time,  or  the  time  when  he  owed  just 
$500,  wa8  96  da.  before  Sep.  16,  or  June  11. 


232 


SIMPLE  INTEREST 


103.  Find  the  equated  time  for  the  payment  of  the  Dr.  side  in 
example  100. 


The  last  debt  was  due  Aug.  2.  Reasoning  as  in  the 
previous  example,  we  find  that  Stone  owed,  Aug.  2, 
$1600  + interest  of  $110,300  for  1  da.,  or  $1600 
+  interest  of  $  1600  for  74  da.  The  equated  timet  or 
tlie  time  wlien  he  owed  only  $  1600,  was  74  days  be- 
fore Aug.  2,  or  May  20. 


Aug.  S.  Dr, 

500  X  14S=  74000 

SOO  X  121  =  S6S00 

700  y.      0-  0 


1500  llOSOO 

1500)110300 
74 


Dr, 

James  Whitman 

Cr. 

1895 

1895 

Apr.  3 

To  Mdse. 

600 

Mayl 

By  Cash 

400 

June  10 

To  Mdae. 

500 

July  15 

By  Cash 

300 

Sep,  15 

To  Mdse. 

400 

Aug.  SO 

By  Cash 

SOO 

104.  Settlement  was  made  ori  the  above  account  Nov.  1,  1895. 
How  much  did  Mr.  Whitman  owe  at  that  time,  counting  interest 
at  6  %  ?  Solve  by  the  interest  method.  Give  the  explanation  in 
full. 

105.  Solve  example  104  by  the  equated  time  method.  Give 
the  explanation  in  full.  Which  do  you  prefer;  the  interest 
method,  or  the  equated  time  method? 

106.  In  the  above  account,  find  the  equated  time  for  the  pay- 
ment of  the  Dr.  side ;  that  is,  find  the  time  when  all  of  the  Dr. 
side  may  be  regarded  as  due. 

107.  In  the  above  account,  find  the  equated  time  for  the  pay- 
ment of  the  Cr.  side ;  that  is,  find  the  time  when  all  of  the  Cr. 
side  may  be  regarded  as  due. 

108.  In  the  above  account,  find  the  interest  on  the  sum  of  the 
debts  from  the  equated  time  to  the  date  of  settlement,  Nov.  1 ; 
find  the  interest  on  the  sum  of  the  payments  to  the  date  of  settle- 
ment ;  find  the  difference. 

Note.—  Ex.  108  illustrates  another  method  of  solving  such  problems. 


INTEREST  233 

ANNUAL   INTEREST 

Annual  interest  is  rarely,  if  ever,  computed  in  actual  business. 
Hie  principal  bears  interest,  and  the  interest  on  the  pnncipal  at 
the  end  of  each  year  hears  interest.     See  p.  216. 

109.  What  is  the  annual  interest  of  $  100  for  3  yr.  3  mo.  3  da. 
at  6%? 

The  simple  interest  of  $  100  for  3  yr.  3  mo.  3  da. 


yr.  MO.  da. 

96    2    3    S 


at  6%  is  a  19.56. 

The  9  6,  interest  on  the  principal  at  the  end  of 


96     1     3     3  the  first  year,  bears  interest  for  2  yr.  3  mo.  3  da. 

96  3     3  the  9  6,  interest  on  the  principal  at  the  end  of  the 

— second  year,  bears  interest  for  1  yr.  3  mo.  3  da. ; 

96     3     9     9  tlie  $ 0,  interest  on  the  principal  at  the  end  of  the 

9 19.55=int.  onnnn.  ^^"^^  y^^^^  bears  interest  for  3  mo.  3  da.    The 

^  -  ,_  .  .  whole  is  equivalent  to  the  interest  of  $6  for  the 

J^3b  —  tnt.  on  int.  ^^^^^  ^^  ^^j^^  intervals,  or  for  3  yr.  J)  mo.  9  da. 

20.91 =annual  int.  '^^^^  interest  of  ^6  for  3  yr.  9  mo.  9  da.  at  6% 

is  9 1.36  ;  the  annual  interest,  9  20.91. 

What  is  the: 

110.   Annual  interest  of  $  420  for  3  yr.  2  mo.  5  da.  @  6% ? 

HI.  Annual  interest  of  $  240  for  1  yr.  6  mo.  12  da.  @  9%? 

112.  Annual  interest  of  $  186  for  4  yr.  9  mo.  18  da.  @  6%? 

113.  Annual  interest  of  $  252  for  5  yr.  5  mo.  15  da.  @  8%? 

114.  Find  the  amount  of  a  note  for  $  1500,  interest  6%,  pay- 
able annually,  given  Sep.  3, 1893,  and  not  paid  until  July  1,  1897. 

115.  Find  the  amount  of  a  note  for  $  2000,  interest  10%,  pay- 
able annually,  given  Jan.  16, 1895,  and  not  paid  until  April  1, 1900. 

116.  Find  the  amount  of  a  note  for  $  600,  interest  8%,  payable 
annually,  given  Oct.  10,  1896,  and  not  paid  until  May  16,  1902. 

117.  Find  the  amount  of  a  note  for  $  1200,  interest  7%,  pay- 
able annually,  given  May  4, 1897,  and  not  paid  until  Feb.  19, 1899. 


234 


COMPOUND  INTEREST 


COMPOUND   INTEREST 

In  all  states,  both  law  and  business  usage  are  opposed  to  the 
collection  of  compound  interest.  There  are  many  problems,  how- 
ever, especially  in  connection  with  insurance,  annuities,  and 
reserve  funds,  where  this  computation  is  necessary. 

Tlie  principal  bears  interest^  the  interest  on  the  principal  at  the  end 
of  each  year  bears  interest^  and  all  other  interest  at  the  end  of  each 
year  bears  interest.    See  p.  ^16. 

lia  What  is  the  amount  at  compound  interest  of  $  1  for  3  yr. 
at  6%, 

During  the  first  year  f  1  is  on  interest ;  it  amounts  to  f  1.06. 

During  the  second  year  $  1.06  is  on  interest ;  since  9 1  amounts  to  $  1.06, 
$  l.Ot)  will  amount  to  1.06  times  8  1.06,  or  ^  (1.06)2. 

During  the  third  year  $  (1.06)-  is  on  interest ;  since  $  1  amounts  to  $  1.06, 
I  (l.OO)'J  will  amount  to  (1.06)^  times  «  1.06,  or$(1.06)«,  or  $1.191016. 

119.  What  is  the  amount  at  compound  interest  of  $  1  for  4  yr. 
at  G%  ?     Explain  as  in  example  118. 


Amocnt  of  $  1  AT  Compound  Interest 


Ykars 

r.% 

6% 

10% 

3 

(1.06)«  =  1.167625 

(1.06)»  =  1.191016 

(1.10)8  =  1.331000 

4 

6 

120.  Explain  as  in  example  118  and  fill  out  the  blanks  in  the 
above  diagram. 

121.  Prove  that  the  amount  of  $1  for  n  years  at  r%  is 
$(l+r)*. 

122.  What  is  the  amount  at  compound  interest  of  $  1  for  2000 
yr.  at  6%  ?  Express  the  answer  as  suggested  in  Ex.  121.  How 
many  decimal  places  are  there  in  the  exact  answer  ? 


INTEREST  235 

123.   Using  the  table  just  prepared,  find  the  compound  interest 
of  $  129.36  for  3  yr.  3  rao.  3  da.  @  6%. 

$(106)^  or  $1   in  3  yr.  at  6%  will  amount  to 

ij.'moje,  anU.il,  Syr.  e(1.06)»,   or  $1.191016;    $  I2».3rt   will 

1X9.36  amount  to   129.36  times  $1.191016,  or 

154.07,  amt.  $129.36  ^^^'^l'        ,     ,^  ^.r.r.„- 

1  nir^n  «.«#  A  1    «  «,«    <?  ^«  At  the  end  of  3  yr.,  $  154.07  is  on  m- 

1.0155,  ami.  9 1,  3  mo,  8  da.  ^   ,      „  ,  .»    ,        ^^   . 

--,  ,^        .It*.  terest  for  3  mo.  and  3  da.    $1  m  this 

156.46,  amt.  whole  time  ^.  .„  ^   .     A,nirr     Ate-*  at 

jgnZf.  time  will  amount  to  $1.0155;  $154.07 

will  amount  to   154.07   times  $1.0155, 

or  $  156.46. 


f  7. 10  int.  whole  time 


124.  In  solving  example  123,  would  it  not  be  better  to  multiply 
$1.0155  by  1.191016  to  find  the  amount  of  $  1  for  the  whole 
time,  and  to  multiply  this  product  by  129.36  ? 

Aru.  No ;  because  the  product  of  the  first  two  is  a  very  long  decimal. 

125.  Without  the  table,  find  the  amount  of  $2630  for  3  yr. 
4  mo.  4  da.  at  4%,  compound  interest. 

Using  the  tal)le  on  p.  236,  find  the  : 

126.  Compound  int.  of  $365.25  for  4  yr.  5  mo.  5  da.  at  5%. 

127.  Compound  int.  of  $  762.28  for  3  yr.  3  mo.  3  da.  at  10%. 
12a  Compound  int.  of  $  625.50  for  5  yr.  6  mo.  4  da.  at  9%. 

Without  the  table,  find  the: 

129.  Compound  int.  of  $  72.38  for  3  yr.  7  mo.  2  da.  at  7%. 

130.  Compound  int.  of  $65.96  for  3  yr.  2  mo.  18  da.  at  8%. 

131.  Compound  int.  of  $85.50  for  1  yr.  9  mo.  16  da.  at  6%. 

Using  the  table,  find  the  : 

132.  Compound  amt.  of  $455  for  10  yr.  6  mo.  10  da.  at  2^%. 
laa  Compound  amt.  of  $  366  for  15  yr.  9  mo.  20  da.  at  4^%. 
134.   Compound  amt.  of  $930  for  18  yr.  4  mo.  4  da.  at  3i%. 


236 


COMPOUND  INTEREST 


Amount  op 

f  1  AT  Compound  Interest  from  1  to  20  Years 

YkaK8 

5«% 

n% 

8% 

84% 

4% 

4*% 

1 

1.0200000 

1.0260000 

1.0300000 

1.0350000 

1.0400000 

1.0460000 

2 

1.0404000 

1.0506260 

1.0609000 

1.0712250 

1.0816000 

1.0926250 

3 

1.0(312080 

1.0768JK)6 

1.0927270 

1.1087178 

1.1248640 

1.14110()1 

4 

1.0824321 

1.1038128 

1.1256088 

1.1475230 

1.1698586 

1.192518(5 

6 

1.1040808 

1.1314082 

1.1692740 

1.1876863 

1.2166529 

1.2461819 

6 

1.1261024 

1.1596934 

1.1940523 

1.2292553 

1.2653190 

1.3022(501 

7 

1.1486866 

1.1886867 

1.2298738 

1.2722792 

1.3159317 

1.3608618 

8 

1.1716693 

1.2184029 

1.2(567700 

1.3168090 

1.3685690 

1.4221006 

0 

1.1960926 

1.2488629 

1.3047731 

1.3628973 

1.4233118 

1.4860951 

10 

1.2189944 

1.2800845 

1.3439163 

1.4106987 

1.4802442 

1.5529694 

11 

1.2433743 

1.3120866 

1.3842338 

1.4599697 

1.5394640 

1.6228530 

12 

1.2682417 

1.3448888 

1.4267608 

1.5110686 

1.6010322 

1.(5958814 

13 

1.2936066 

1.3785110 

1.4686337 

1.5(539560 

1.6660735 

1.7721961 

14 

1.3194787 

1.4129738 

1.6125897 

1.6186945 

1.7316764 

1.8519449 

15 

1.3458683 

1.4482981 

1.5579674 

1.6753488 

1.8009435 

1.9352824 

16 

1.3727857 

1.4846056 

1.6047064 

1.7339860 

1.8729812 

2.0223701 

17 

1.4002414 

1.6216182 

1.6628476 

1.7946756 

1.9479(X)5 

2.1133768 

18 

1.4282462 

1.6596587 

1.7024330 

1.8674892 

2.0258165 

2.2084787 

1» 

1.4668111 

1.6986501 

1.7535060 

1.9225013 

2.1068491 

2.3078603 

20 

1.4859474 

1.638(3164 

1.8061112 

1.0897888 

2.1911231 

2.4117140 

Years 

r>% 

6% 

'% 

8% 

9% 

10% 

1 

1.0500000 

1.0600000 

1.0700000 

1.0800000 

1.0900000 

1.1000000 

2 

1.1025000 

1.1236000 

1.1449000 

1.16(54000 

1.1881000 

1.2100000 

3 

1.1576250 

1.1910100 

1.2250430 

1.2597120 

1.2950290 

1..33lO(KiO 

4 

1.2155063 

1.2(524770 

1.3107960 

1.3604890 

1.4115816 

1.4041000 

6 

1.2762816 

1.3382256 

1.4025517 

1.4693281 

1.5386240 

1.6105100 

6 

1.3400956 

1.4185191 

1.5007304 

1.5868743 

1.6771001 

1.7715610 

7 

1.4071004 

1.503(5:i03 

1.6057815 

1.7138243 

1.8280391 

1.9487171 

8 

1.4774554 

1.6938481 

1.7181862 

1.8509302 

1.9925626 

2.1435888 

9 

1.5513282 

1.6894790 

1.8384592 

1.9990046 

2.1718933 

2.3579477 

10 

1.6288946 

1.7908477 

1.9671514 

2.1589250 

2.3673(537 

2.5937425 

11 

1.7103394 

1.8982986 

2.1048520 

2.331(5390 

2.58042(;4 

2.8531167 

12 

1.7958563 

2.0121965 

2.252101G 

2.5181701 

2.812(5648 

3.1384284 

13 

1.8856491 

2.1329283 

2.4098450 

2.719(5237 

3.0658046 

3.4522712 

14 

1.9799316 

2.2609040 

2.5785342 

2.9371936 

3.3417270 

3.7974983 

16 

2.0789282 

2.3965582 

2.7590315 

3.1721691 

3.6424825 

4.1772482 

16 

2.1828746 

2.5403517 

2.95216,38 

3.4259426 

3.9703059 

4.5949730 

17 

2.2920183 

2.6927728 

3.1588152 

3.7000181 

4.3276.334 

5.0544703 

18 

2.4006192 

2.8543392 

3.3790323 

3.99(50195 

4.7171204 

5.5599173 

19 

2.5269502 

3.0255995 

.3.6165275 

4.3157011 

5.141(;013 

6.1159090 

20 

2.6532977 

3.2071355 

3.8696845 

4.6609571 

5.6044108 

6.7275000 

INTEREST  237 

From  the  tablCj  the  amount  for  any  time  may  he  found. 
135.   What  is  the  ainoiint  of  $  100  for  90  years  at  6%  com- 
pound interest? 

S3.i^071355,  amt.  $  1,  20  yr.  «3.20  is  on  interest  at  the  end 

oon'Ti^f-r  of20yr.     During  the  next  20  yr., 

iS.^UiliSOo  ^  J  ^^.jjj  amount  to  $3.20  ;  «3.20+ 

f  10.2857179,  amt.  ^i,  40  yr.  will  amount  to  3.20+  times  $3.20, 

10.2857179  or  $10.28. 

$10.28  18  on   interest   at  the 


f  105.7959927 y  ami.  $1,  80  yr.  end  of  40  yr.     During  the  next 

1.7908477  ^0  y^»  '^ ^  ^^'^'^  amount  to  $  10.28; 

— ^— — — ^^^^— ^^  etc. 

9 189.4645099,  amt.^l,  90  yr.  '^i^^   pupil  should   finish   the 

f  18940. 45,  amt.  $100  explanation. 

If  interest  is  payable  at  intervals  of  less  than  a  year,  the 
example  should  be  reduced  to  an  equivalent  example,  interest 
payable  annually. 

136.  What  is  the  amount  at  compound  interest,  of  $  100  for 
3  yr.  2  mo.  4  da.  at  8%,  interest  payable  quarterly  ? 

^irr    Smo       Ada.  ^^^^"^    quarterly    is    paying 
4                           of  a  year.     The  interest  for  a 

12  yr.  8  mo.  16  da.  P^"^  °"^  ^^^^^^  ®^  *  ^^'^^  ^  ^% 

is  the  same  as  the  interest  for  a 

whole  year  @  2%;  .*.  the  interest  @  8%,  payable  quarterly,  is  the  same 

as  the  interest  for  4  times  3  yr.  2  mo.  4  da.,  or  for  12  yr.  8  mo.  16  da.  @  2  %, 

payable  annually. 

The  pupil  should  finish  the  explanation. 

137.  Show  that  the  amount  for  5  years  at  12%,  payable  semi- 
annually, is  the  same  as  the  amount  for  2  times  5  years,  at  \  of 
12%  payable  annually. 

13a  Find  the  amount  at  compound  interest,  of  $  100  for  'Jo  yr. 
6  mo.  7  da.  at  4%. 

139.  Find  the  amount  at  compound  interest,  of  $  100  for  8  yr. 
3  mo.  2  da.  at  12%,  interest  payable  3  times  a  year. 


INVOLUTION   AND   EVOLUTION 


INVOLUTION— TERMS  AND  RELATIONS 

8  =  2  X  2  X  2,  or  8  =  2^ 
If  the  product  is  wanting,  this  becomes 

what  =  2^? 
It  means  "  what  is  the  product  when  2  is  used  3  times  as  a 
factor?''    Seep.  146, 

Involution  is  the  process  of         ^. 
o    J-        .V  J     i.      t         ^1  What  18  the  product  when  2  is 

hnding  the  product  when   the      used  3  times  as  a  factor  ? 

same  number  is  used   several  ,       u    o     o     o     o 

^ns.    o;  z  X  ^  X  ^  ^  o. 
times  as  a  factor. 

The  result  is  the  power.  ^  ^  '^'  third  pouter  of  2. 

1.  Read:  5^  5^  5^  5'. 

Ans.   6^,  5  square,  or  5  to  the  second  power ;  6',  6  cube,  or  6  to  the  third 
power ;  6*,  6  to  the  fourth  power ;  6*,  5  to  the  a;th  power. 

2.  Write   the   squares  of  the  integers    from   13  to  25,   and 
memorize  the  results. 

3.  Write  the  cubes  of  the  integers  from  1  to  10,  and  memorize 
the  results. 

4.  Declare  the  results  rapidly:   16^  25^  13^;  17^;  21";  22^: 
24*;  142;  152.  182.  192;  20*;  23*;  9«;   7^;  2»;  4»;  6»;  3^  5^  8^. 

To  raise  a  factor  to  any  power,  write  the  base,  and  over  it,  the 
product  of  the  exponent  by  the  number  denoting  the  required  power. 
Illustration :   (48)2  ^  46  .  (48)2  =:  4*  x  4*  =  (4  x  4  x  4)  x  (4  x  4  x  4)  =  46. 

5.  Find  the  value  of:  (2«/;  (2^)3;  (3*)^  (Sy. 

e.   The  square  of  12  is  144 ;  what  is  its  6th  power  ?      Ans.  144^ 
7.   The  cube  of  12  is  1728 ;  what  is  its  6th  power  ?     Ans.  1728*. 


INVOLUTION  AND  EVOLUTION  239 

EVOLUTION  —  TERMS   AND   RELATIONS 

2»  =  8. 
If  the  base  is  wanting,  this  becomes 

(what)'  =  8  ?,  what  =  VS  ?,  or  what  =  S'  ?. 
It  means  "what  number  must  be  taken  3  times  as  a  factor 
to  produce  8  ?  " 

Evolution  is  the  process  of  find-  Tm.  .    •    .i            v        u-  t. 

,.  .                 ,  What    IS  the    niimber  which 

ing   the   number   which   must   be  must  be  taken  3  times  as  a  factor 

taken  several  times  as  a  factor  to  to  produce  8  ? 

produce  a  eiven  product.  -'^'**'  '^^  because  ^  =  8. 

The  result  is  the  roo^  2  is  the  tkirU  root  of  S, 

a  Read:  V9;  -5^;  ^/Sl;  -^81. 

Ans.  VO,  the  square  root  of  9,  or  the  second  root  of  9 ;  v^,  the  cube 
root  of  27,  or  the  tliird  root  of  27  ;  \/8T,  the  fourth  root  of  81 ;  \/8T,  the  xth 
root  of  81. 

9.  Read:  9*;  27^;  81^  81*. 
Ans.  These  are  read  as  in  Ex.  8.    Also,  9*  may  be  read,  9  to  the  }  power ; 
27%  27  to  the  }  power  ;  .  .  .  . 

10.  Declare  the  results  rapidly :  V225;  ViOO;  V625  ;  Vl69; 
V324;    VTUG;    ^/^56 ;    V^SO;    V3(U ;    V441 ;    V484 ;    V529. 

U.  Declare  the  results  rapidly:  "v/m;  \/27;  \/l26;  ^3^; 
■\/S;  </U'y  </2l6;  ■v^512. 

A  root  of  a  perfect  power  may  be  extracted  by  factoring. 

By  factoring  J  find  the : 

12.  Value  of  V5184  15.  Value  of  ^248832 

la   Value  of  V\72S  16.   Value  of  \/41()0625 

14.  Value  of  \/32768  17.   Value  of  \/7962624 

Ex.  12.     Ana.  V6184  =  VO*  x  b-'  =  9  x  8  =  72. 

Ex.  15.    Ans.   v^248,832  =  v^9»  x  8»  x  6  =  v'S*  x  2»  x2»  =  3x2x2=12. 


240 


SQUARE   ROOT 
SQUARE   ROOT 


1.  If  a  number  is  separated  into  periods  of  two  figures  eachy 
beginning  with  units'  place^  the  number  of  peiiods  will  be  equal  to 
the  number  of  figures  in  the  square  root  of  that  number. 

Illustration 

P  =  1  99^  =  98'01  That  is,  the  square  of  a  number  of 

9^  =  81  100^  =  I'OO'OO  1  figure  has  1  period ;  of  2  figures,  2 

10^  =  100  999^  =  99'80'01  periods ;  of  3  figures,  3  periods ;  etc. 


II.  If  a  number  is  separated  into  periods  of  two  figures  each, 
beginning  at  units'  place^  the  square  ivot  of  the  number  denoted  by 
the  left-hand  period ^  will  give  the  first  figure  of  the  root;  the  square 
root  of  the  number  denoted  by  the  first  ttco  periods,  will  give  the  first 
two  figures  of  the  root ;  etc. 


Illustratioh 


ltS4^  =  l'5g*S7»56 
P=l 

1^  =  I'M 
ISS^  =  1'51'29 


or 


\/V5S'S7'56  =  12S4 

V7  =  /,  first  figure 

y/¥52  =  1£+,  first  two  fig. 

y/1'52'27  =  123 -^y  first  three  fig. 


The  number  denoted  by  the  first  period  is  1;  its  square  root  gives  the  first 
figure  of  the  root. 

The  number  denoted  by  the  first  two  periods  is  152 ;  its  square  root  gives 
the  first  two  figures  of  the  root ;  etc. 


III.    (a-\-by  =  a'-\-2ab-^b%      or      a'+(2a  +  b)b 


a  +  b 
a  +  b 


a^  +  ab 
a^  +  2  a6  +  6* 


In  a^  +  2ab-\-  b^ 
6  is  a  factor  of  the  last  two  terms. 
2ab-^b  =  2a;  b^  ^  b  =  b.     There- 
fore, a2  +  2  aft  +  62= 

a2  +  (2aH-6)6 


INVOLUTION  AND  EVOLUTION  241 

la  Extract  the  square  root  of  625. 

(a  +  6)«  =  a«  +  2  a6  +  6^ 
=  a2  +  (2  a  +  6)6 


€'25(25 
4 

6i25{25 
4 

40 

5 
45 

225           a  =  20 
b=   5 

225 

45    225 

225 

We  separate  the  number  into  periods  of  two  figures  each,  because  the 
square  root  of  the  first  period  will  give  the  first  figure  of  the  root,  and  the 
square  root  of  the  first  two  periods  will  give  the  first  two  figures  of  the  root. 

We  raise  a  +  6  to  the  second  power  to  have  a  perfect  square  before  us, 
and  factor  the  terms  containing  6  to  cause  the  second  term  of  the  root  to 
appear. 

To  obtain  a,  the  first  term  of  the  root  in  the  model,  we  must  extract  the 
square  root  of  a'.  Hence,  to  obtain  the  first  term  of  the  root  in  this  ex- 
ample, we  must  extract  the  square  root  of  what  corresponds  to  a'',  or  of  6. 
Ve  =  2  ;  a  —  20,  because  G  is  not  0  units,  but  6  hundreds. 

To  obtain  6,  the  second  term  of  the  root  in  the  model,  we  must  divide  2  ab 
by  2  a.  Hence,  to  obtain  the  second  term  of  the  root  in  this  example,  we 
must  divide  what  corresponds  to  2  ab,  or  the  greater  part  of  226,  by  what 
corresponds  to  2  o,  or  by  40.    226  h-  40  =  5  ;  6  =  6. 

To  obtain  the  rest  of  the  model,  we  must  multiply  what  is  within  the 
parenthesis,  that  is,  (2a  +  6),  by  6.  Hence,  to  obtain  the  rest  of  this  num- 
ber, we  must  multiply  what  corresponds  to  what  is  within  the  parenthesis,  by 
what  corresponds  to  6.  6  =  6;2a  +  6=45;46x6  =  226.  Since  there  is 
no  remainder,  ^626  =  26. 

Note.  —  Instead  of  writing  40  as  at  the  left,  it  is  customary  to  write  4,  regard- 
hig  it  as  4  tens.    The  5  may  then  be  added  without  an  extra  line. 


Extract  the  square  root  and  explain  the  process  : 

19.  9025.  22.   1369.  25.  8281. 

2a  9409.  23.   2304.  26.  1024. 

21.  6889.  24.   7225.  27.  532a 

AMKK.   AKITII.  — 16 


242 


SQUARE   ROOT 


2a   Extract  the  square  root  of  1 '52 '27 '56. 

We  separate  the  number  into  periods  of  two 
figures  each,  because  the  square  root  of  the  first 
period  will  give  the  first  figure  of  the  root ;  tlie 
square  root  of  the  first  two  periods,  the  first 
two  figures ;  etc. 

FIRST 

We  extract  the  square  root  of  1'62  to  obtain 
tlie  first  two  figures  of  the  root.  Tlie  explana- 
tion is  the  same  as  already  given.  The  pupil 
should  give  it  in  full.     Seep.  SU- 


SECOND 

We  extract  the  square  root  of  1'62'27  to  ob- 
tain the  first  three  figures  of  the  root.  We  may 
regard  152'27  as  152  hundreds  and  27  units. 
That  is,  we  are  again  to  extract  the  s(|uare  root 
of  a  number  of  two  periods,  the  first  being  152  ; 
the  second,  27. 

To  obtain  a,  the  first  term  of  the  root  in  the 
model,  we  must  extract  the  square  root  of  a^. 
Hence,  to  obtain  the  first  term  of  the  root  in 
this  example,  we  must  extract  the  square  root  of 
what  corresponds  to  a^  or  of  152.  v/l52  =  12  ; 
a  =  120,  because  152  is  not  162  units  but  162 
hundreds.  The  pupil  should  finish  the  expla- 
nation. 

THIBD 

We  extract  the  square  root  of  1'52'27'66  to 
obtain  the  first  four  figures  of  the  root.  We 
may  regard  1522 7 '56  as  16227  hundreds  and  56 
units.  That  is,  we  are  again  to  extract  the 
square  root  of  a  number  of  two  periods,  the 
first  period  being  15227  ;  the  second,  56. 

V15227  =  123  ;  a  =  1230,  because  15227  is 
not  15227  units  but  15227  hundreds.  The  pupil 
should  finish  the  explanation. 

Note.  —  In  practice,  only  the  third  form  should 
be  written. 


(a-\'b)^=a^-\-2ab-\-b^ 
=  a^-^(2a  +  b)t 


VSX'trse 
1 


[12 


92 


SECOND 

l'5g'27'56 

1 


\12S 


22 


24s 


827 

729 

98 


THIRD 


1'52'27'66\12S4 

1 


22 


243 
2464 


827 
729 


9856 
9856 


INVOLUTION    AND   EVOLUTION  243 

Find  the  value  of: 


29. 

Vy85y(5                   31.    V390625 

33.    V1079521 

30. 

V53361                  32.    V522729 

34.   V3345241 

35. 

Extract  the  square  root  of  J|. 

1^5  _  5                         We  extract  the  square  root  of  the  nuinei 
^S6~~6                    ^r,  and  then  of  the  denominator. 

aa  Extract  the  square  root  of  .000025. 

/  QQiQQiag  __  QQg    fQy.  We  point  off  into  periods  of  two  figures 

*        '  ^  each,  beginning  at  the  decimal  point. 

/  fifu)nc2K  —  ^  /       ^^  ^®  point  off  as  many  decimal  places  in 

~~  \  iOOOOOO      *'^®  ^^^^  ^  there  are  decimal  periods  in  the 
y-  number. 

There  are  3  decimal  periods  in  the  num- 


1000  ber,  and  3  decimal  places  in  the  root. 

37.  Extract  the  square  root  of  2 


y/2  =  ^2.'00W00+  =  14U^- 

3a  Extract  the  square  root  of  .00365. 

/  nnt9^'rn  Caution.  —  Be  sure  to  point  off,  beginning 

V.CW  3b  50  ^i^jj  ^j,e  decimal  point. 

39.  Extract  the  square  root  of  j. 

V^  —  -v/  ^R'RR'RfiJi.'  ^^  ^^*^  denominator  is  not  a  perfect  square, 

"g—^-^^  ^^  "^ "''  it  is  best  to  reduce  the  fraction  to  a  decimal. 

Find  the  value  of: 

4a    r^)*  43.    (J)i  46.    (f)i 


41.    V:0625  44.   V6:25  47.   V.0()00626 


42.   V.625  45.   V62:6  4a   V.00U626 


244  CUBE  ROOT 

CUBE  ROOT 

I.  If  a  number  is  separated  into  periods  of  three  figures  each, 
beginning  with  units'  place,  the  number  of  jyeriods  will  be  equal  to  the 
number  of  figures  in  the  cube  root  of  that  number. 

Illustration 

i«  =        1      99*=        970' 299  That  Is,  the  cube  of  a  number  of 

9»  =     7S9     10(fi  =      VOiM/OOO  1  figure  has  1  period  ;  of  2  figures,  2 

10*  =  I'OOO     999*  =  997'OOg  999  periods  ;  of  3  figures,  3  periods,  etc. 

II.  If  a  number  is  separated  into  periods  of  three  figures  each, 
beginning  with  units'  place,  the  cube  root  of  the  number  denoted  by 
the  left  hand  period,  ivill  give  the  first  figure  of  the  root;  the  cube  root 
of  the  number  denoted  by  the  first  two  periods,  will  give  the  first  two 
figures  of  the  root,  etc, 

12S4*  =  l'879f080'904  ^''    y/l'879'080'904  =  1284 

i«  =  /  v^  =  /,  first  figure 

Ifg*  =  l'7t8  y/1'879  =  12  ^,  first  two  figures 

128*  =         1' 860' 867  y/l'879'080  =128-\-,  first  three  figures 

The  number  denoted  by  the  first  period  is  1 ;  its  cube  root  gives  the  first 
figure  of  the  root. 

The  number  denoted  by  the  first  two  periods  is  1879 ;  its  cube  root  gives 
the  first  two  figures  of  the  root ;  etc. 

III.  (a  -f  bf  =  a''  +  3a«6  +  3a6*  +  6»,  or  a»  +(3a*  +  Sab  +  b')b 
a  -\-b 


aft  +  6*  6  is  a  factor  of  the  last  three  terms. 

a«  +  2a6  +  6»  Za^b^b  =  Sa';    Sab'' -^  b  =  Sab; 

a^h  6»  -T-  6  =  62.    Therefore, 

a^  +  2a^b-\-    ab^  a*  +  Sa^b  +  Sab^+ b*  = 

a^b  +  2ab'^  +  b*  a*  +  (S a^  ■{■  S ab  +  b^)b 

a»  +  3a2&  +  3a62  +  6» 


INVOLUTION  AND  EVOLUTION 


245 


49.   Extractthecuberoot  of  1728. 


(a  +  6)'  =  a»  +  Sa^b  +  Sab^  +  6« 
=  a8  +  (3a«  +  3a6+62)6 


i 

i 

'728{12 

r 

soo 

728    a- 

10 

60 

6  = 

2 

S64 

728 

We  separate  the  number  into  periods  of  three  figures  each,  because  the 
cube  root  of  the  first  period  will  give  the  first  figure  of  the  root,  and  the  cube 
root  of  the  first  two  periods  will  give  the  first  two  figures  of  the  root. 

We  raise  «  +  6  to  the  third  power  to  have  a  perfect  cube  before  us,  and 
factor  the  terms  containing  h  to  cause  the  second  term  of  the  root  to  appear. 

To  obtain  a,  the  first  term  of  the  root  in  the  model,  we  must  extract  the 
cube  root  of  a".  Hence,  to  obtain  the  fii-st  term  of  the  root  in  this  example, 
we  must  extract  the  cube  root  of  what  corresponds  to  a',  or  of  1.  y/\  —  \\ 
a  =  10,  because  1  is  not  1  unit,  but  1  thousaiid. 

To  obtain  ft,  the  second  term  of  the  root  in  the  model,  we  must  divide 
Sa^b  by  'da'K  Hence,  to  obtain  the  second  term  of  the  root  in  this  example, 
we  must  divide  what  corresponds  to  .Sa^ft,  or  the  greater  part  of  728,  by  what 
corresponds  to  3  ««,  or  by  .300.     728  h-  300  =  2 ;  b  =  2. 

To  obtain  the  rest  of  the  model,  we  must  multiply  what  is  within  the 
parenthesis,  that  is  (3a^  +  3a6  4-  b^)y  by  6.  Hence,  to  obtain  the  rest  of 
this  number,  we  must  multiply  what  corresponds  to  what  is  within  the  paren- 
thesis, by  what  corresponds  to  6.  3  a6  =  60 ;  fo'^  =  4  ;  3  a^  +  3  aft  +  ft'  .=  3«4  ; 
36^  X  2^  728.    Since  there  is  no  remainder,  v^l728  =  12. 

Extract  the  ctibe  root  and  explain  the  process : 

50.  12167.  53.  405224.  56.   7049G9. 

51.  39304.  54.  314432.  57.   148877. 

52.  117649.  55.   250047.  5a   166375. 


Note. —The  pupil  should  observe  that  the  explanation  is  identical  with  that 
for  the  extraction  of  8i|iiare  ri>ot.  In  each  case,  the  Icailing  of  the  formula  is 
followed.    Compare  with  p.  241. 


246 


CUBE   ROOT 


59.  Extract  the  cube  root  of 
l'879'()80'i)04. 

We  separate  the  number  into  periods 
of  three  figures  each,  because  the  cube 
root  of  the  first  period  will  give  the  first 
figure  of  the  root ;  the  cube  root  of  the 
first  two  periods,  the  first  two  figures,  etc. 

FIRST 

We  extract  the  cube  root  of  1'879  to 
jbtain  the  first  two  figures  of  the  root. 
The  explanation  is  the  same  as  already 
given.  ITie  pupil  should  give  it  in  full. 
See  p.  S45. 

^       ^  SECOND 

We  extract  the  cube  root  of  1'879'080  to 
obtain  the  first  three  figures  of  the  root. 
We  may  regard  1879'080  as  1879  thou- 
sands and  080  u  nits.  That  is,  we  are  again 
to  extract  the  cube  root  of  a  number  of 
two  periods,  the  first  being  1879;  the 
second,  080. 

To  obtain  a,  the  first  term  of  the  root 
in  the  model,  we  must  extract  the  cube 
root  of  a*.  Hence,  to  obtain  the  first  term 
of  the  root  in  this  example,  we  must  ex- 
tract the  cube  root  of  what  corresponds  to 
a»,  or  of  1879.  v'l879=12  ;  a  =  120,  be- 
cause 1879  is  not  1879  units,  but  1879 
thousands.  The  pupil  should  finish  the 
explanation.         ^,„„^ 

^  THIRD 

We  extract  the  cube  root  of  1  '870'080'904 
to  obtain  the  first  four  figures  of  the  root. 
We  may  regard  1879080'904  as  1879080 
thousands  and  904  units.  That  is,  we 
are  again  to  extract  the  cube  root  of  a 
number  of  two  periods,  the  first  period 
being  1879080  ;  the  second,  904. 

v^l'879'080  =  123  ;  a  =  12.30,  because 
1879080  is  not  1879080  units  but  1879080 
thousands.  The  pupil  should  finish  the 
explanation. 

NoTB.  —  In  practice  only  the  tliird 


(a  +  6)»  =  «»  +  3a«6  +  3a6«  -}-  b^ 
=  a»  +  (3aH3a6+62)6 

FIRST 

V879'080'm{lt 

1 

SOO  \S79 
60 
4 
364 


a  =  10 
b  =  g 


728 
151 

SBCOITD 

V879'08O'9ok{liS 

1 


SOO 

879 

60 

4 

364 

728 

43200 

151080 

1080 

9 

44^89 

1328tj7 

a  =  10 
b  =  2 


a  =  120 
b  =  3 


18213 

THIRD 

1'879'080'B04{12S4 

1 


a  =  10 
b  =  2 


300 

60 

4 

364 

879 
728 

43200 

1080 

9 

151080 

44289 

132867 

4538700 

14760 

16 

18213904 

4553471 

S 

18213904 

a  =  120 
b  =  3 


a  =  1230 
h  =  4 


form  should  be  written. 


INVOLUTION  AND  EVOLUTION  247 

Find  the  vcUue  of: 

ea   ^94818816  62.   ^1067462648 

61.   ^177504328  63.    </i039B09i97 

64.  Extract  the  cube  root  of  f}|. 

sfl^S     5  W®  extract  the  cube  root  of  the  numerator, 

Sl'oJZ  ~  ^  and  then  of  the  denominator. 

65.  Extract  the  cube  root  of  .000125. 

•i/  000' 125  =  05    far  ^®  P°"**  °^  ^^*°  periods  of  three  figures 

*  '     *  ^  each,  beginning  at  the  decimal  point. 

3/  ooQiinr  _   3/     125  We  point  off  as  many  decimal  places  in 

•y/.UtfU         ~~\ 2000000  ^^e  root  as  there  are  decimal  periods  in  the 

^  number. 
=  -^  There  are  2  decimal  periods  in  the  number 

^00  and  2  decimal  places  in  the  root. 

66.  Extract  the  cube  root  of  2. 
</^  =  y/2:000'000'000-\'=1.259+ 

67.  Extract  the  cube  root  of  .00365. 

V/vig>/?/r^  Caution.  —  Be  sure  to  point  off,  beginning 

V  006  oou  ^j^j^  ^jjg  decimal  point. 

6a  Extract  the  cube  root  of  }. 

«S  __  t/ -, If  ^^®  denominator  is  not  a  perfect  cube, 

\^  ""  'S/.666  666 -\-  n  is  best  to  reduce  the  fraction  to  a  decimal. 

FHnd  the  value  of: 

69.    (H)^  72.    (J)*  75.    (f)i 

7a  ^.001728  7a  -s/vm  7a  </.ooooi728 

71.   <^.0001728  74.    \^i72:8  77.   </.0000r6625 


248  ANY  ROOT 

Ainr  ROOT 

To  extract  any  root  of  a  number j  point  off  into  periods  of  as  many 
figures  each  as  there  are  units  in  tfie  index  of  the  root,  raise  a  +  b 
to  the  corresponding  power y  and  proceed  as  the  formula  indicates. 

7a  Extract  the  4th  root  of  2847396321. 

(o  +  by  =  a*  +  4a»6  +  6a^b^  +  4a6»  +  b* 
=  o«  +  (4a3  +  6a26  +  4a6*^  +  6»)6 

2S*47S9'6S21i2Sl 
16 


S2000 

7200 

726 

27 
S9947 

1247S9 
119841 

a  =  20 
b  =  S 

4866800 

S174a 

92 

4898632 

0 
0 
0 
1 
~1 

48986S21 
48986321 

a  =  230 
b  =  l 

Note.  — The  pupil  should  supply  the  explanation. 

79.  l*repare  the  formula  for  tlie  extraction  of  the  fifth  root. 

80.  Prepare  the  formula  for  the  extraction  of  the  sixth  root. 
Fi7id  the  

81.  Value  of  <^221o33456.  83.  Value  of  ^9354951841. 

82.  Value  of  ^7902624.  84.  Value  of  -^2985984. 

To  depress  a  factor  to  any  root,  ivrite  the  base,  and  over  it  the 
quotient  of  the  exponent  by  the  number  denoting  the  required  root. 
Illustration :  V^  —  5^ ;  because  5^  x  o'*  x  5'^  =  5«. 

85.  Find  the  value  of:  v^^  W^',  VS^;  \/5^. 

86.  Find  the  value  of:  -^/V64;  -\/\/64;  -J/64. 

Ans.  ■V^V8i  =  2;  V^  =  2  ;   \/6i  =  2  ;  Vv^  =  ^/v^  =  \/x. 

87.  Solve  Ex.  78  by  extracting  the  square  root  of  the  square 
root. 


MENSUKATION 


We  may  consider  that  which  has  no  dimension,  that  which  has 
one  dimension,  that  which  has  two  dimensions,  or  that  which  has 
three  dimensions. 


NO  DIMENSION 

That  which  has   no   dimension  is  a 

point. 

NoTK.  —  Strictly  speakin;;,  the  illustrati(»n  is 
not  a  point,  because  however  small  it  may  be,  it 
has  length,  breadth,  and  thickness. 


Illustrations 
Point 


ONE  DIMENSION 

That  which  has  one  dimension  is  a 
line. 

A  line  may  extend  in  the  same  direc- 
tion, a  straight  line;  or  it  may  con- 
stantly change  its  direction,  a  curved 
line. 

If  two  straight  lines  in  the  same  plane 
are  extended,  they  will  meet,  or  they 
will  not  meet.  If  they  do  not  meet, 
they  are  parallel;  if  they  meet,  they 
form  angles. 

S49 


Lines. 
Straight.  2.  Curted. 


PanUlel 


/ 


Angiat. 


260 


ONE   DIMENSION 


ONE   DIMENSION 

If  two  lines  meet,  the  angles  will 
be  equal,  right  angles ;  or  not  equal, 
oblique  angles ;  the  larger  is  obtitsej 
the  smaller  acute. 

A  straight  line  may  be  parallel  to 
the  horizon,  a  horizontal  line;  per- 
pendicular to  the  horizon,  a  vertical 
line ;  or  neither  parallel  nor  perpen- 
dicular to  tlie  horizon,  an  oblique 
line. 


/ 


IAS. 
SA4. 


ABflM. 

Right.  8.  Obtuse. 

Oblique.  4.  Acute. 


/ 


Draw  and  define : 

1. 

A  point. 

2. 

A  line. 

a 

A  straight  line. 

4. 

A  curved  line. 

5. 

Parallel  lines. 

6. 

Two  perpendicular  lines. 

7. 

A  vertical  line. 

15 

16. 

19. 


20. 


1.  Horizontal  Line. 

2.  Vertical  Line. 
8.  Oblique  Line. 


8.  A  horizontal  line. 

9.  An  oblique  line. 
10.  An  angle. 

U.  A  right  angle. 

12.  An  obtuse  angle. 

13.  An  acute  angle. 

14.  Two  oblique  angles. 

17.  la 


•/  \- 


21. 


22. 


/■     \ 


Draw  points  and  lines  situated  like  the  above,  and  from  each 
point  draw  a  perpendicular  to  the  nearest  line. 


MENSURATION 


251 


TWO  DIMENSIONS 


That  which  has  two  dimensions  is  a  sur- 
face. 

If  any  two  points  of  a  surface  are  con- 
nected by  a  straight  line,  that  line  will  lie 
wholly  on  the  surface,  a  pZane  sin-face^  or 
plane;  or  it  will  not  lie  wholly  on  the  sur- 
face, a  curved  surface. 

If  straight  lines  inclose  a  surface,  the 
figure  is  a  j)dygon. 

The  least  number  of  straight  lines  which 
can  inclose  a  plane  is  three,  a  triangle^  (1). 
The  three  lines  may  be  equal,  an  equilateral 
triangle,  (2)  ;  two  of  them  may  be  equal,  an 
isosceles  triangle,  (3) ;  or  no  two  of  them  may 
be  equal,  a  scalene  triangle,  (4). 

A  triangle  may  have  one  right  angle,  a 
rigid-angled  triangle,  (6)  ;  one  obtuse  angle, 
an  obtuse-angled  triangle,  (7) ;  or  three  acute 
angles,  an  acute-angled  triangle,  (8). 

The  next  number  of  straight  lines  which 
can  inclose  a  plane  is  four,  a  quadrilateral, 
(9).  The  quadrilateral  may  have  both 
pairs  of  its  opposite  sides  parallel,  B.paral- 
lelograyn,  (10) ;  one  pair  parallel,  a  trapezoid, 
(11);  or  neither  pair  parallel,  a  trapezium, 
(12). 

A  parallelogram  may  have  its  angles 
right  angles,  a  rectangle,  (13) ;  or  not  right 
angles,  a  rhomboid,  (14). 

The  rectangle  may  have  its  sides  all 
equal,  a  square,  (15);  the  rhomboid  may 
have  its  sides  all  eipial,  a  rhombus,  (16). 


The  bee  of  this  pa^e  Is  a 
plane  surfkce,  ur  a  plaoe. 


Curved  surface. 

Triangles. 
2.   EquilatercU. 
8.   Inoftcele*. 
4.  Scalene. 

K 


% 


Triangles. 
Right-angled. 
Obtuae-ang.'ed. 
AouU-angltd. 


\ 


Quatlrilaterab. 

10.  ParaiUlogram. 

11.  TVapetoid. 

12.  Trapezium. 
18.  JtecUtngU. 

14.  Rhomboid. 

15.  Square. 
1«. 


252 


TWO   DIMENSIONS 


Five  sides  may  inclose  a  surface,  pentagon;  six 
sides,  hexagon;  seven  sides,  heptagon;  eight  sides, 
octagon;  nine  sides,  now agfon ;  ten  sides,  cfeccw/o/i, .... 

A  polygon  may  have  its  sides  and  angles  equal, 
a  regular  polygon;  or  its  sides  and  angles  not 
equal,  an  irregular  polygon.  Hence  we  may  have 
regular  and  irregular  pentagons,  regular  and 
irregular  hexagons  .... 

A  regular  polygon  of  an  infinite  number  of 
sides  is  a  circle. 


Regular  bexuf;i>n. 
Irregular  hexagon. 


Cizvle. 


Define : 

23.  A  surface. 

24.  A  polygon. 

25.  A  triangle. 

26.  An  equilateral  triangle. 

27.  An  isosceles  triangle. 
2a  A  scalene  triangle. 

29.  A  right-angled  triangle. 


30.  An  acute-angled  triangle. 

31.  A  regular  polygon. 

32.  A  regular  pentagon. 

33.  A  regular  hexagon. 

34.  A  regular  heptagon. 

35.  An  obtuse-angled  triangle. 

36.  A  circle. 


37.  Beginning  with  ^^ plane  surface^^^  (see  Note)  define :  paral- 
lelogram; rectangle;  rhomboid;  rhombus;  square. 

3a  Beginning  with  quajdrilateraJy  (see  Note)  define :  parallelo- 
gram; rectangle;  square;  rhombus. 

Note.  — A  definition  may  bejjin  with  different  terms,  e.g.: 

A  square  is  a  plane  surface  bounded  by  two  pairs  of  opposite  sides,  having: 
each  pair  parallel,  having  its  angles  all  right  angles,  and  having  its  sides  all 
equal.    Or, 

A  square  is  a  quadrilateral  having  each  pair  of  its  opposite  sides  parallel,  hav- 
ing its  angles  all  right  angles,  and  having  its  sides  all  equal.    Or, 

A  square  is  a  parallelogram  having  its  angles  all  right  angles,  and  having  its 
sides  all  equal.    Or, 

A  square  is  a  rectangle,  having  its  sides  all  equal. 

That  definition  is  the  best  which  is  tlie  shortest,  provided  it  begins  with  a  term 
which  is  understood  by  the  person  for  vchom  the  definition  is  given. 


MENSURATION 


253 


PARTS   OF  A  POLYGON 


That  part  of  a  polygon  on 
which  it  is  supposed  to  rest  is 
its  base;  the  distance  around  a 
polygon,  its  perimeter;  the  perim- 
eter of  a  circle,  its  circiirnference. 

In  a  right-angled  triangle,  the 
side  opposite  the  right  angle  is 
the  hypotenuse;  the  other  sides, 
legs. 


BC,  Bufte. 
AB+BC-^AC,  PeHmeUr. 


AB  and  BC,  Leg». 
AC,  l/ypotenune. 


The    altitude    of    a    triangle, 
parallelogram,  or  trapezoid,  is  a 
perpendicular  to  the  base  from         d    d 
the  vertex  opposite  the  base. 


AB  is  the  altitude  in  each  of 
these  figures.  Observe  that  the 
base  must  sometimes  be  extended. 

Since  a  triangle  may  be  re- 
garded as  resting  on  any  one 
of  its  sides,  it  may  have  three 
bases  with  an  altitude  corre- 
sponding to  each. 

The  apothem  of  a  regular  poly- 
gon is  the  perpendicular  from 
its  center  to  any  side. 

The  diagonal  of  a  polygon  is 
a  line  which  joins  any  two  ver- 
tices not  adjacent 


h^. 


Due  d        b  X 

AB,  Altitude. 


Bate,  BC;  AUitude,  AD. 
Base.  AB;  Altitude,  CH. 
Ba-,  AC;  Altitude,  BE. 


(D 


ABy  Apothem. 


^ 


BD,  DUtffonal. 


254 


TWO   DIMENSIONS 


PARTS  OF  A  CIRCLE 


A  circle  is  a  plane  figure  bounded  by  a  curved  line,  every  point 
of  which  is  equally  distant  from  a  point  within  called  the  center. 


LiNKAR  Parts 

We  may  consider  the  whole  of  the 
bounding  line,  the  circumference;  or  a 
part  of  it,  an  arc. 

A  line  may  cut  the  circumference  in 
two  points,  a  secant;  or  may  touch  it 
at  one  point,  a  tangent. 

A  line  may  be  drawn  from  the  center 
to  any  point  of  the  circumference,  a 
radius. 

A  line  may  connect  any  two  points 
of  the  circumference,  a  chord.  A  chord 
may  pass  through  the  center,  a  diameter. 


M     a 

Linear  parts. 

ABEFGD,  Circuwprene^. 
DG,  Arc. 
HM,  Secant. 
ST,  Tangent. 
CB,  Radiue. 
AB,  Chord. 
DB,  Diameter. 


Sdrfacb  Parts 

We  may  consider  the  portion  of  a 
circle  between  a  chord  and  its  arc,  a 
segment;  or  the  portion  between  two 
radii  and  their  arc,  a  sector.  A  sector 
may  be  half  of  the  circle,  a  semicircle; 
a  quarter,  a  quadrant ;  a  sixth,  a  sextant. 


Surfitce  parta^ 

ABX,  Segment. 
BCF,  Sector. 
DOE,  Semicircle. 
EGC,  Quadrant. 
CFG,  Sextant. 


Draw  and  define : 
39.   A  circle. 
4a   A  segment. 

41.  A  sector. 

42.  A  quadrant. 
4a   A  sextant. 


44.  A  semicircle. 

45.  A  radius. 

46.  A  chord. 

47.  A  diameter. 
4a   A  tangent. 


49.  A  secant. 

50.  An  arc. 

51.  The  center. 

52.  A  circum- 

ference. 


MENSURATION 


255 


COMPUTATIONS  —  LINEAR   PARTS 

I.  The  circumference  of  a  circle  is  equal  to  twice  the  radius  times 
S.W6. 

II.  The  square  of  the  hypotenuse  of  a  right-angled  triangle  w 
equod  to  the  sum  of  the  squares  of  the  other  two  sides. 

Note.  —  For  illustration  and  explanation,  the  pupil  should  turn  to  p.  272. 

S3.  The  radius  of  a  circle  is 
5  in. ;  find  its  circumference. 
BelatioH :  C  =  2  x  R  x  3.  I4I6 
C=  S  X  5x  3.I4I6  =  SI.4I6 
Circumference  =  SI.4I6  in. 


We  Bubstitata  the  valae  given. 


54.  The  circumference  of  a  circle  is  31.416  in. ;  find  its  radius. 

Relation :  C  =  £  x  R  x  3. 14I6 

SI.4I6-2  X  R  X  3.14I6 

^^     SI.4I6  ^^ 

We  sabstitate  the  ralae  giren. 


2  X  S.I4I6 
Radius  =  6  in. 


The  pnpll  should  always  draw  the  fl^re. 


55.  The  hypotenuse  of  a  right-angled  triangle  is  10  in. ;  its 
perpendicular  6  in. ;  find  its  base. 

Relation :  AB^  =  AC^  +  BC^ 

100  =  36+  Bd^ 

100-36=  BCf^  =  64 

BC  =  V64  =  8  C 

Base  =  8  in. 

56.  A  room  is  20  ft.  x  12  ft.  x  9  ft.    How  far  is  it  from  a  lower 
corner  to  the  opposite  upper  corner  ?  d 

A^=144  +  4O0  =  5U  ^/9 

DC*  =  81  +  644  =  625 

DC  =  V62S  =  25 

Distance  =  25  ft. 

A  C,  Diagonal  <if floor;  A  D,  //eight  of  room. 


266  LINEAR   PARTS 

Find  the  circumference :  Find  the  radius  : 

57.   Radius  8  in.  60.   Circumference  41.888  ft. 

sa   Radius  6  in.  61.   Circumference  502.G56  in. 

59.   Diameter  20  ft  62.  Circumference  15.708  m. 

Find  the  hypotenuse :  Find  the  other  leg  : 

63.  Legs  6  in.  and  8  in.  67.  Hypot.  40  in. ;  base  32  in. 

64.  Legs  24  in.  and  10  in.  6a  Hypot.  13  in.;  base  12 in. 

65.  Legs  32  in.  and  24  in.  69.  Hypot.  30  in. ;  base  18  in. 

66.  Legs  12  in.  and  16  in.  70.  Hypot.  35  in. ;  base  21  in. 

71.  If  the  foot  of  a  ladder  50  ft.  long  is  put  30  ft.  from  the  base 
of  a  wall,  how  far  will  it  reach  ?     Draw  the  figure. 

72.  A  ladder  91  ft.  long  was  placed  between  two  buildings. 
The  base  being  at  the  same  point,  the  ladder  reached  a  point  84 
ft.  from  the  ground  on  the  first  building,  and  35  ft.  from  the 
ground  on  the  other.  How  far  apart  were  the  two  buildings? 
Draw  the  figure. 

73.  A  room  is  30  ft  long,  18  ft  wide,  and  13J  ft.  high.  How 
far  is  it  from  a  lower  corner  to  the  opposite  upper  corner  ?  Draw 
the  figure. 

74.  What  is  the  length  of  the  longest  straight  rod  which,  with- 
out bending,  can  be  put  into  a  box  5  ft  long,  1  yd.  wide,  and  J  yd. 
deep  ?     Draw  the  figure. 

75.  Two  boats  start  from  the  same  point  and  sail,  one  north 
10,560  ft.,  the  other  east  7920  ft     How  far  apart  are  they  at  last  ? 

76.  Two  towers,  94  and  78  ft  high,  are  situated  on  opposite 
banks  of  a  river  30  ft.  broad.  What  is  the  length  of  the  shortest 
line  connecting  the  tops  of  the  towers  ?     Draw. 

77.  Two  objects,  A  and  B,  are  in  a  straight  line  south  of  a  flag- 
staff 16  ft  high.  If  the  lines  joining  the  top  of  the  flagstaff  with 
each  object  are  20  ft.  and  34  ft.  respectively,  how  far  apart  are  A 
and  B  ?     Draw  the  figure. 


MENSURATfON  257 

COMPUTATION  —  AREAS 

The  area  of  a  polygon  is  the  number  of  square  units  in  its 
surface.  The  square  unit  is  one  of  the  denominations  in  square 
measure. 

NoTK.  —  For  illustrations,  explanations,  and  proofs,  the  pupil  should  turn  to 
p.  121,  and  to  pp.  272,  273,  and  274. 

III.  Tlie  area  of  a  rectangle  is  equal  to  the  product  of  its  base 
by  its  altitude. 

IV.  The  area  of  a  parallelogram  is  equal  to  the  product  of  its 
base  by  its  altitude. 

V.  The  area  of  a  triangle  is  equal  to  one  half  the  product  of  its 
base  by  its  altitude. 

VI.  Tlie  area  of  a  triangle  is  equal  to  the  square  root  of  the  con- 
tinued product  of  the  half  sum  of  its  sides  and  the  remainders  found 
by  subtracting  each  side  from  the  half  sum  separaldy. 

VII.  The  area  of  a  trapezoid  is  equal  to  one  half  the  product  of 
the  sum  of  its  parallel  sides  by  its  altitude. 

VIII.  TTie  area  of  a  regular  polygon  is  equal  to  one  half  the 
product  of  its  perimeter  by  its  apothem. 

IX.  T7ie  area  of  a  circle  is  equal  to  the  square  of  its  radius 
times  S.I4I6. 

X.  The  area  of  an  irregular  polygon  is  equal  to  the  sum  of  the 
areas  of  the  triangles  into  which  it  may  be  divided. 

7a  The  area  of  a  triangle  is  12  sq.  ft. ;  its  altitude  6  ft.  Find 
its  base. 

Relation :  Area  =  ^  B  x  A, 
B  =  T  =  ^. 

j5^j^  —  t  fl  We  tutMtltuto  Uie  value  jfivou. 

AMBR.    ARITH.— 17 


258 


AREAS 


79.  Find  the  area  of  a  triangle  whose  sides  are  6  in.,  8  in.,  and 
10  in. 


8  —  \  sum  of  Ihe  sides. 
a^h,^  c—  the  sides  respectively. 


Relation :  C  =  2  x  i2  x  3. 1416 
C=Sx  5  X  3.1416  =  31.416 
Circumference  =  31.416  in. 


lielation:  Area=  V»(»— a)(»— 6)(»  — c) 

Area  =  y/lS  x6x4^t  =  ^4 

Area  =  S4  SQ-  in. 

80.  The  area  of  a  circle  is  78.54  sq.  in.     Find  its  circumference. 

lielation :  Area  =  R^  x  3. 14 16 
78.54  =  R^x  3.1416 

3.1416 
R  =  5in. 

81.  The  area  of  an  equilateral 
triangle  is  62.352  sq.  rd.  Find 
its  perimeter. 

Relation :  Area  =  '-BCx  AD 

=  -x.866x  =  .4S3^ 
62,352  =  . 433  x^ 

.4SS         ^ 
z=lt 
Perimeter  =  36  rd.  ^ 

82.  Find  the  area  of  a  trapezium  whose  sides  are  9,  15,  13,  and 
14  m.  respectively,  and  whose  shortest  diagonal  is  12  m. 

c 
A lea  =  area  ABC  +  area  ACD. 

Tlie  pupil  should  complete  the 
work. 

83.  The  circumference  of  a  circle  is  125.664  m.     Find  the  area 
of  an  inscribed  square.  ^     ^\-4 

Relation:  C  =  2  x  R  >(  3.1416 
125.064  =  2  X  R  X  3.1416 

B  =     ^^^'^^^   =  20 


AD  =  —  =  .866x 


?x  3.14I6 
AB  =  40 


1600 
800 
800  sq  m 


mp:nsufiation  259 

84.  Find  the  base  of  a  rectangle  whose  area  is  48  A.  and  alti- 
tude 48  rd. 

85    Find  the  area  of  a  triangle  whose  sides  are  25  rd.,  60  rd. 
and  65  rd. 

85.  Find  the  area  of  a  square  whose  diagonal  is  20  m. 

87.  Find  the  diagonal  of  a  square  whose  area  is  625  sq.  ft. 
8a   Find  the  area  of  a  trapezoid  whose  parallel  sides  are  60 
rd.  and  80  rd.,  and  whose  altitude  is  30  rd. 

89.  Find  the  area  of  a  rhombus,  one  of  whose  sides  is  12  dm 
and  altitude  8  dm. 

90.  Find  the  altitude  of  a  rhombus  whose  area  is  48  sq.  rd. 
and  base  48  rd. 

91.  Find  the  area  of  a  triangle  whose  base  is  9  m,  and  altitude 
6  m. 

92.  Find  the  area  of  a  trapezium  whose  diagonal  is  40  rd.,  and 
perpendiculars  from  the  opposite  vertices,  16  rd.  and  20  rd. 

93.  Find  the  area  of  a  circle  whose  diameter  is  20  m. 

94.  Find  the  area  of  a  regular  octagon,  one  of  whose  sides  is 
8  feet,  and  whose  apothem  is  9.656  ft. 

95.  Find  the  altitude  of  a  triangle  whose  area  is  48  A.  and 
base  48  rd. 

96.  Find  tlie  circumference  of  a  circle  whose  area  is  392.70 
sq.  rd. 

97.  Find  the  altitude  and  area  of  an  equilateral  triangle,  each 
of  whose  sides  is  20  ft. 

9a  Find  the  altitude  and  area  of  an  isosceles  triangle  whose 
base  is  40  ft.  and  whose  equal  sides  are  52  ft. 

99.   What  is  the  area  of  a  square  circumscribed  about  a  circle 
whose  circumference  is  314.16  ft.  ? 

100.  A  circular  ring  is  formed  by  two  circles  having  the  same 
center.  The  radius  of  the  inner  circle  is  8  ft;  the  ratlins  of  the 
outer  circle  is  10  ft.  Draw  the  circular  ring,  and  compute  its  area. 
What  is  the  ratio  of  the  circumferences  of  the  bounding  circles  ? 


260 


THREE   DIMENSIONS 


THREE   DIMENSIONS 


That  which  has  three  dimensions 
is  a  solid. 

That  part  on  which  a  solid  rests  is 
its  base;  its  other  surfaces  sltb  faces; 
the  union  of  two  faces  is  an  edge; 
the  union  of  three  or  more  edges,  a 
vertex. 

A  solid  may  have  two  bases  equal 
and  parallel  polygons,  and  its  faces 
rectangles,  a  prism.  If  its  bases  are 
triangles,  triangular  prism;  squares, 
square  prism ;  ....  circles,  circidar 
prism,  or  cylinder. 

A  solid  may  have  two  bases  parallel 
polygons,  and  its  faces  trapezoids, 
frustum  of  a  pyramid.  If  its  bases 
are  triangles, /rw^^Mm  of  a  triangular 
pyramid;  ....  circles,  frustum  of  a 
circular  pyramid,  or  frustum  of  a  cone. 

A  solid  may  have  one  base  and  its 
faces  triangles,  a  pyramid.  If  its 
base  is  a  triangle,  triangular  pyra- 
mid ;  square,  square  pyramid ;  circle, 
circular  pyramid,  or  cone. 

A  solid  bounded  by  four  surfaces 
is  a  tetrahedron;  eight  surfaces,  an 
octahedron  ;  twenty  surfaces,  an  icosa- 
hedron;  six  squares,  a  cube;  twelve 
surfaces,  a  dodecahedron ;  a  curved 
surface  every  point  of  which  is 
equally  distant  from  the  center,  a 
sphere. 


Illustrations 

A 


A  BCD,  Solid. 
BCD,  Bate. 
A  CD,  Foci: 
AC,  Edge. 
At   Vertex. 


A,  //exagonal  prism. 

B,  Cylinder. 


5 


A,  Fruetum  of  hexagonal  pyramid. 

B,  Frustum  of  cone. 


i  4 

A,  Bemagonal  pyramid. 

B,  Cone. 


A,   Tetrahedron.  B,  Cube. 

C,  Icoeahedron.  D,  Sphere. 


MENSURATION 


261 


The  entire  aurfcice  of  a  solid  is  the 
number  of  square  units  in  all  its  sur- 
face. 

The  convex  surface  of  a  solid  is  all 
its  surface  but  the  bases. 

The  volume  of  a  solid  is  the  num- 
ber of  cubic  units  in  its  contents. 

The  altitude  of  a  cone  or  pyramid 
is  the  perpendicular  distance  from  a 
vertex  to  the  plane  of  the  base ;  the 
altitude  of  a  cylinder,  prism,  or  frus- 
tum of  a  pyramid  or  cone  is  the  per- 
pendicular distance  between  its  bases. 

The  slant  height  of  a  regular  pyra- 
mid is  the  perpendicular  distance 
from  its  vertex  to  one  of  the  sides  of 
its  base. 


n 


Square  onlt.  Cubic  unit. 


The  entire  surikoe  la  ABC+ ACD 
+  ABD  +  BCD. 

The  convex  sur&ce  Is  ABC+  ACD 
+ABD. 

The  altitude  \s  AE;  AKia  peri)en- 
dicular  to  BDC. 

The  slant  height  is  A II;  the  base  of 
the  face  A  CD  is  the  line  CD;  All  is 
perpendicular  to  CD. 


Define : 

101.   Solid. 

116. 

Square  pyramid. 

102.   Base  of  a  solid. 

117. 

Pentagonal  pyramid. 

103.   Face  of  a  solid. 

ua 

Hexagonal  pyramid. 

104.   Edge  of  a  solid. 

119. 

Cone. 

105.   Vertex  of  a  solid. 

12a 

Tetrahedron. 

106.   Prism. 

121. 

Octahedron. 

107.   Triangular  prism. 

122. 

Icosahedron. 

lOa    S(iuare  prism. 

123. 

Hexahedron. 

109.   Pentagonal  prism. 

124. 

Dodecahedron. 

HO.   Hexagonal  prism. 

125. 

Sphere. 

HI.  Cylinder. 

126. 

Altitude  of  a  pyramid. 

112.   Frustum  of  a  pyramid. 

127. 

Slant  height. 

113.   Frustum  of  a  cono. 

12a 

Convex  surface  of  a  solid. 

U4.   Pyramid. 

129. 

Entire  surface  of  a  solid. 

115.  Triangular  pyramid. 

130. 

Volume  of  a  solid. 

262 


CONSl  RUCTIONS 


CONSTRUCTIONS 

The  pupil  should  make  solids  from  pasteboard.  Each  form 
should  be  cut  out  from  a  single  piece.  It  should  then  be  folded 
and  pasted. 

131.  Construct  a  triangular  prism. 

Draw  two  parallfl  lines.  Lay  off  on  each 
as  many  equal  distances  as  the  base  is  to 
have  sides.  Conni'ct  tlie  points  of  division. 
Construct  upon  two  opposite  sides,  &8  AB 
and  DC^  regular  polygons.  Prepare  flaps 
for  pasting  as  represented  by  the  dotted 
lines.  Cut  entirely  through  outside  lines 
and  partly  through  inside  lines.  Fold,  and 
paste  the  flaps  on  the  inside. 

To  DRAW  THK  Eqi;ilateral  Triangle. 
With  A  and  B  as  centers  and  a  radius 
equal  to  AB,  draw  arcs  of  circles.  These 
arcs  will  intersect  at  the  vertex  of  the  tri- 
angle ;  draw  AO  and  OB. 

132.  Construct  a  cylinder. 

Proceed  as  in  drawing  the  form  for  a 
prism  ;  prepare  the  flap,  roll  over  until  AD 
coincides  with  BC,  and  paste  on  the  inside. 

Note.  —  No  base  is  necessary.  Light  card- 
board or  paper  should  be  used. 

133.  Construct  a  cone. 

From  any  point  as  a  center  draw  an  arc 
of  a  circle.  Lay  off  any  convenient  dis- 
tance, as  J5C;  draw  AD  for  the  flap.  Cut 
entirely  through  the  outside  lines ;  roll  the 
form  until  AB  coincides  with  ^C,  and  paste 
the  flap  on  the  inside. 

Note.  —  No  base  is  necessary.  Light  card- 
board or  paper  should  be  used. 

134.  Construct  a  square  prism.     See  Ex.  187, 

135.  Construct  an  hexagonal  prism.     See  Ex.  138. 


MKNSU  RATION 

136.   Construct  a  frustum  of  a  cone. 

Proceed  as  in  drawing  the  form  for  a  cone, 
constructing  two  arcs  from  the  same  center. 
Cut  out  liB'D'D,  roil  over  until  BB'  coincides 
with  CC^  and  paste  on  the  inside. 


2m 


137.   Construct  a  square  pyramid. 

Proceed  as  in  drawing  tlie  form  for  a  cone. 
From  B  lay  off  as  many  equal  distances  as  the 
base  i.s  to  have  sides.  Connect  the  points  of 
division  with  A.  Construct  on  one  of  the  sides, 
as  EF^  a  regular  polygon.  Prepare  flaps  for 
pasting  as  represented  by  dotted  lines.  Cut  en- 
tirely through  outside  lines  and  partly  through 
inside  lines.  Fold,  and  paste  the  flaps  on  the 
inside. 

To  nitAw  THE  Square.  Prolong  EF,  and 
lay  off  FII  equal  to  EF.  From  E  and  H  as 
centers,  with  a  radius  greater  than  EF,  draw 
arcs  of  circles  and  connect  the  points  of  inter- 
section. Lay  off  FO  equal  to  EF.  From  0 
and  E  as  centers,  with  a  radius  equal  to  EF, 
draw  arcs  of  circles ;  connect  their  intersection 
with  O  and  E. 


13a  Construct  a  frustum  of  an  hex- 
agonal pyramid. 

Proceed  as  in  drawing  the  form  for  a  frus- 
tum of  a  cone.  I>ay  off  on  BD  and  B'D'  as 
for  a  pyramid.  Construct  on  two  sides,  as  EF 
and  E'F*,  regular  polygons.  Prepare  flaps, 
cut  and  paste  as  before. 

To  i>RAW  THE  Regular  Hexagon.  From 
E  and  F  as  centers,  draw  arcs  of  circles  with 
a  radius  equal  to  EF.  From  0,  'heir  point  of 
intersection,  as  a  center,  with  a  radius  equal 
to  EF,  describe  a  cin^le.  From  F,  lay  off  on 
tiie  circumference,  distances  equal  to  EF\  con- 
««ect  the  points  thua  found. 


E F 


4-4 


204  CONVEX  SURFACES 

COMPUTATIONS  — CONVEX   SURFACES 

XI.  The  convex  surface  of  a  prism  is  the  product  of  the  perimeter 
nf  its  base  by  its  altitude. 

XII.  The  convex  surface  of  a  cylinder  is  the  product  of  the  cir- 
cumference of  its  base  by  its  altitude. 

XIII.  Tlie  convex  surface  of  a  pyramid  is  half  the  product  of  the 
perimeter  of  its  base  by  its  slant  height. 

X I  \\  The  convex  surface  of  a  cone  is  half  the  product  of  the  cir- 
cumference of  its  base  by  its  slant  height. 

XV.  The  convex  surface  of  a  frustum  of  a  pyramid  is  half  the 
product  of  the  sum  of  the  perimeters  of  its  two  bases  by  its  slant 
height. 

XVI.  7%e  convex  surface  of  a  frustum  of  a  cone  is  half  the 
product  of  the  sum  of  the  circumferences  of  its  two  bosses  by  its 
slant  height. 

XVII.  The  surface  of  a  sphere  is  four  times  the  square  of  its 
radius  times  3.1410- 

Note.  — For  illustrations,  explanations,  and  proofs,  the  pupil  should  turn  to 
p.  2<>1,  and  to  pp.  275  and  276. 

139.  Find  the  convex  surface  of  a  cylinder,  radius  of  the  base 
6  in.,  altitude  8  in. 

Belation:  S=  Cir.  x  Alt. 

S=gx  Bx  S.I4I6  X  Alt.  Conv.  surf.  =  301.5936  sq.  in. 

=  2x6  X  3.1416  X8  =  301.5936 

140.  Find  the  convex  surface  of  a  square  pyramid,  one  side  of 

the  base  4  in.,  slant  height  12  in. 

Belation :  S=i Per.  x  S.  H. 

Conv.  surf.  —  96  sq.  in. 
S  =  -gX  16  X  12  =  96 

141.  Find  the  convex  surface  of  a  frustum  of  a  cone,  radius  of 
upper  base  6  in.,  radius  of  lower  base  8  in.,  slant  height  12  in. 

Belation :  8  =  ^{0 -h  C')x  S.  R. 
S=l(Sx6xS.1416-\-2x8x3.1416)xl2        Conv.  surf  =  527.7888  sq.  in. 
=  527.7888 


MENSURATION  265 

142.  Find  the  radius  of  a  sphere  whose  surface  is  314.16  sq.  ft. 

Relation:  5  =  4  x  if»  x  S.I4I6 
314-16  =  4  X  if«  X  3.14I6 
jp  -     SI4.I6    _  25  Badiua  =  6  ft. 

4  X  S.I4I6 

NoTB.  — The  relation  is  stated  in  its  natural  form,  and  the  given  terms  are 
« ihstitated  in  the  relation. 

Find  the  convex  surface  of:  Find  the  surface  of: 

143.  A  sq.  pyr. ;   side  of  base,  146.   A  sphere,  whose 

6  in. ;  S.  H.,  4  in.  radius  is  6  in. 

144.  A  cone ;  area  of  base  50.2656  147.  A  sphere,  whose 

sq.  ft. ;  alt,  3  ft.  diameter  is  10  in. 

145.  Acyl.;  radius  of  base,  6  in. ;  14a   A  cube,   whose 

alt.,  4  in.  diagonal  is  12  in. 

Find  the : 

149.  Radius  of  a  sphere  whose  surface  is  201.0624  sq.  in. 

150.  Circum.  of  a  sphere  whose  surface  is  804.2496  sq.  ft 

151.  Edge  of  a  cube  whose  surface  is  864  sq.  ft 

152.  Diagonal  of  a  cube  whose  surface  is  150  sq.  ft. 

153.  Find  the  convex  surface  of  a  frustum  of  a  square  pyramid, 
one  side  of  upper  base  4  ra,  of  lower  base  8  m,  slant  height  24  m. 

154.  Find  the  convex  surface  of  a  frustum  of  a  cone,  radius  of 
upper  base  6  ft,  of  lower  base  11  ft.,  altitude  12  ft 

155.  What  is  the  approximate  area  of  the  earth's  surface,  its 
diameter  being  nearly  8000  mi.  ? 

156.  What  is  the  convex  surface  of  a  rectangular  prism,  base 
8  ft.  X  6  ft,  altitude  10  ft  ? 

157.  Find  the  cost,  at  1^  a  sq.  ft.,  of  painting  a  church  spire 
whose  base  is  a  pentagon,  each  side  6  ft,  and  whose  slant  height 
is  60  ft. 


266  VOLUMES 

COMPUTATIONS  -VOLUMES 

XVIII.  The  volume  of  a  prism  is  equal  to  the  product  of  the  area 
of  its  base  by  its  altitude. 

XIX.  The  volume  of  a  cylinder  is  equai  to  the  product  of  the 
area  of  its  base  by  its  altitude. 

XX.  Tfie  volume  of  a  pyramid  is  equal  to  one  third  the  product 
of  the  area  of  its  base  by  its  altitude. 

XXI.  The  volume  of  a  cone  is  equal  to  one  third  the  product  of 
the  area  of  Us  base  by  its  altitude. 

XXII.  The  volume  of  a  frustum  of  a  pyramid  is  equal  to  one 
third  the  product  of  the  sum  of  the  areas  of  its  upper  base^  lower 
basCj  and  mean  proportional  base,  by  its  altitude. 

XXIII.  Tfie  volume  of  a  frustum  of  a  cone  is  equal  to  one  third 
the  product  of  the  sum  of  the  areas  of  its  upper  base,  loiver  ba>se,  and 
mean  proportional  base,  by  its  altitude. 

XXIV.  The  volume  of  a  sphere  is  equal  to  four  thirds  times  the 
cube  of  its  ra/iins  times  S.IJ^IG. 

Note.  — For  illustrations,  explanations,  and  proofs,  the  papil  shoald  turn  to 
p.  122,  p.  261,  and  pp.  276,  277,  and  278. 

158.  Find  the  volume  of  a  square  prism,  each  side  of  the  base 
4  ft,  altitude  12  ft. 

Relation  :   V=  B  x  Alt.  Volume  =  19S  cu.  ft. 

V=  16x  12=  19S 

159.  Find  the  volume  of  a  triangular  pyramid,  each  side  of  the 
base  4  ft.,  altitude  12  ft. 

Helation :   V  =  ^Bx  Alt.  Volume :  27. 71  cu. ft. 

V=  iV6  X2x2x2x.l2  =  27.71 

s 


MENSURATION 


267 


ISO.   Find  the  volume  of  a  frustum  of  a  cone,  radius  6f  the  upper 
base  6  ft.,  radius  of  the  lower  base  8  ft.,  altitude  12  ft. 

Relation :  V=^iB+B'-\-B")  A  It. 

36  X  3.1416=  B 


64  X  3.1410=  W 

4S  X  3.1416=  B" 
148  X  3. 1416  =  sum  areas 

4 

59S  X  3.1416  =  1859. 827 g 
Volume  =  1859.8272  cu.  in. 

NoTB.  —  For  meaning  of  mean  proportional,  see  p.  171,  ITT. 

Id.  The  volume  of  a  sphere  is  113.0976  cu.  in.    Find  its  surface. 


B"  =  y/36  X  3. 14I6  y.  64  x  3. 14I6 

=  6x8x3.  I4I6 

=  4Sx  3.14I6 

It  is  better  to  extract  the  sq.  rt. 
of  the  factors  before  performing  the 
multiplication. 


Relation:  V  =  ^x  R^  x  3.14I6. 
113.0976  =  ^  X  iJ«  X  3.14I6 


339.2928 

R* 

R 


4x  R^x  3.14I6 
339.2928 


4  X  3.14I6 
3 


27 


Relation :  6'  =  4  x  i2«  x  3.14I6. 

S  =  4x  9x  3.14I6 

Surface  =  113.0976  sq.  in. 


NoTB  —  The  relation  is  stated  in  its  natural  form,  and  the  given  terms  are 
substituted  iu  the  relation. 


Find  the  volume  of: 

162.  A  tri.  prism ;  each  side 
of  base  4  in. ;  alt.  10  in. 

163.  A    cylinder;    radius   of 
base  6  in. ;   alt.  10  in. 

Find  the  required  jmrt : 

166.  The  vol.  cyl.  3141.6  cu. 
in.;  alt.  10  in.;  D  of  bane  f 

167.  The    vol.   cone    188.406 
cu  m  ;  alt.  8  m;  area  of  base? 

168    The  vol.  cone  336  cu  m; 
alt  12  m;  Dofbasef 


164.  A  tri.  pyr. ;  each  side  of 
base  6  in.;  alt.  12  in. 

165.  A    cone;     diameter    of 
base  12  in. ;   alt.  15  in. 

169.  The    vol.   of    a    sphere 
904.7808  cu.  in. ;  Df 

170.  The    vol.   of    a    sphere 
33.5104  cu.  in.;    Sf 

171.  The  vol.  of  a  cube  110592 
cu.  in.;  Sf 


SIMILARITY 


SIMILARITY 

Two  figures  may  be  alike  in  form,  similar  figures.  In  order 
that  figures  may  be  similar,  two  conditions  must  be  fulfilled : 

For  eveini  angle  of  the  one,  there  must  be  a  corresponding  equal 
angle  in  the  other. 

Tlie  aides  about  the  equal  angles  must  be  proportional. 

Similar  Triangles 


Similar  Quadrilaterals 


.1^ 


Similar  Prisms 


:^d 


Similar  Cones 


MENSURATION  269 

XXV.  In  similar  Jigures,   linear  parts  are  to  each  ether  as 
homologous  linear  jyarts. 

XXVI.  In  similar  figures,  surfaces  are  to  ea^h  other  as  the 
squares  of  homologous  linear  parts. 

XXVII.  In  similar  figures,   volumes  are  to  each  other  as  tJie 
cubes  of  homologous  linear  parts. 

172.  In  the  similar  triangles,  find  the  ratio  of  AB  to  ab. 

AB:ah::S:l  Prin.  xxv. 

173.  Find  the  ratio  of  area  triangle  ABC  to  triangle  abc. 

ABC:abc::fi:fi  Prin.  xxvi. 

::9  : 1 

174.  In  the  similar  cones,  find  the  ratio  of  their  surfaces. 

::16:l  Prm.  xxvi. 

175.  In  the  similar  cones,  find  the  ratio  of  their  volumes. 

vol.A:vol.a::S^:£^  Prin.  ixvlL 

.'  .*  64  •'  1 

17&  If  the  volume  of  the  large  prism  is  288  cu.  in.,  what  is  the 
volume  of  the  smaller  ? 

vol  l8t :  vol.  Sd : :  BL^ :  6d*  In  similar  figures,  vol- 

£88 :  vol.  Sd : :  €*:  3^  umes  are  to  each  other  as 

£88  X  S^  ^^^  cubes  of  homologous 

vol.  2d  =  — — —  =  S6  linear  parts. 

177.  The  convex  surface  of  a  sphere  is  100  sq.  ft.    What  is  the 
convex  surface  of  a  sphere  whose  radius  is  twice  as  long  ? 

LoNO  Way  Short  Wat 

Let  X  =  radius  small  sphere  Since  the  radius  of  the 

ix  =  radius  large  sphere  larger  is  twice  the  radius 

sur.  small :  sur.  large : :  x' :  (2xy  of  the  smaller,  the  surface 

100 :  sur.  large : :  x-* :  4  x*  of  the  larger  must  be  the 

sur  larae  -  ^^  ^  ^'^^  -  AM  Square  of  2,  or  4  times  the 

^  ^  ^^  smaUer,  or  400  sq.  ft. 


270  SIMILARITY 

178.  If  a  pipe  2  mches  in  diameter  discharges  40  gal.  per  min- 
ute, how  much  will  a  pipe  3  inches  in  diameter  discharge  ? 

Section  Ist  .section  Sd  : :  S* :  ^,  or  'n.e  amount  of  water  discharged 

40:  discharge  M  ::  4  '  9  depends  upon  the  area  of  a  trans- 

discharge  2d  =  ^^7^  =  90  verse  section  ;  this  section  is  a  circle. 

179.  If  it  costs  $1200  to  build  a  house  20  ft.  by  30  ft.,  what 

will  be  the  approximate  cost  of  a  house  the  same  height,  40  ft. 

by  60  ft.  ? 

^    .  .  .        _.  -^       -^    -^     .^    ^r»  Since  the  heights  are  the  same, 

CoBt  1st  :  cost  gd  : :  tOxSO  .'40x60       ^,,^  ,,^^^^^  ^^  „^,  ^j„,i,^^^    ,^,j^^  ^^^^^ 


::  1:4 


depends  approximately  upon  tlie  area 


Rs  :  whole  : . 

■  P 

:ifi 

.-. 

'  1  . 

'4 

B'8=l  off 2 

=  50f 

A's  =  fL50 

CostSd  =  f4S00  of  the  base 

18a  A  and  B  buy  a  grindstone  4  in.  thick  and  2  ft.  in  diameter. 
A  uses  the  stone  until  the  diameter  of  the  part  left  is  1  ft.  If 
the  whole  stone  cost  $  2,  how  much  ought  each  to  pay  ? 

The  whole  and  B's  part  are  not 
similar  because  they  have  the  same 
thickness.  The  cost  of  each  part  de- 
pends upon  the  area  of  the  base. 

All  problems  under  Similarity  may  be  solved  in  other  ways ;  in 
some  cases,  it  is  best  to  neglect  the  rules  on  p.  269. 

181.  Find  the  radius  of  a  circle  having  an  area  equal  to  the 
sum  of  the  areas  of  two  circles  whose  radii  are  3  in.  and  4  in. 

Relation:  area  of  a  circle  =  R^  x  S.14I6. 
Areal8t  =  9x  S.14I6 

Area  Sd  =16  x  S.I4I6  Note.  — In  many  cases,  it  is  wise  to  de- 
Sum  =  25  X  3.I4I6  fer  multiplying  and  dividing  as  lonj;  as  pas- 
Sum  =  R^x  S.I4I6  sible.    S«e  p.  72.    See,  also,  Ex.  mo,  p.  267. 
— -                ^^  In  this  example,  it  would  be  unwise  to 
.-.  R^  X  3.1416  =  26X3.  I4I6  multiply  \)  by  3.U16. 
R^  =  25 
R  =  5  in. 


MENSURATION  271 

182.  The  volume  of  a  cone  wliose  altitude  is  8  in.,  is  50.2656  cu. 
in.  What  is  the  volume  of  a  similar  cone  whose  altitude  is 
12  in.? 

18a  The  convex  surface  of  a  fiiistum  of  a  pyramid  whose 
altitude  is  12  in.  is  432  sq.  in.  What  is  the  altitude  of  a  similar 
frustum  whose  convex  surface  is  3888  sq.  in.  ? 

184.  The  area  of  a  circle  is  10  sq.  in.  What  is  the  area  of  a 
circle  whose  diameter  is  twice  the  diameter  of  the  first  ? 

185.  Two  lead  pipes  are  respectively  1  inch  and  2  inches  in 
diameter.  The  area  of  a  horizontal  section  of  the  larger  is  how 
many  times  a  similar  section  of  the  smaller  ? 

186.  How  many  lead  pipes  1  inch  in  diameter,  will  discharge  as 
much  water  as  one  pipe  4  inches  in  diameter? 

187.  A  cannon  ball  weighs  32  lb.  What  is  the  weight  of  a 
similar  ball  whose  diameter  is  half  the  diameter  of  the  first  ? 

18a   What  is  the  ratio  of  the  surface  of  the  two  balls  in  Ex.  187  ? 

189.  A  is  6  ft.  tall;  his  bronze  statue  is  12  ft.  tall.  If  the 
length  of  A's  little  finger  is  2J  in.,  what  is  the  length  of  the  little 
finger  of  the  statue  ? 

190.  If  it  costs  $  1  to  paint  a  statue  of  A's  size,  how  much  will 
it  cost  to  paint  the  statue  in  Ex.  189  ? 

191.  If  a  statue  of  A's  size  weighs  500  lb.,  how  much  will  the 
statue  in  Ex.  189  weigh  ? 

192.  If  a  bin  6  ft.  deep  holds  60  bu.,  what  are  the  contents 
of  a  similar  bin  12  ft.  deep? 

193.  The  volume  of  a  sphere  is  100  cu.  ft.  What  is  the  volume 
of  a  sphere  whose  surface  is  10  times  as  great  ? 

194.  Four  pipes,  each  2  inches  in  diameter,  empty  into  a  tank. 
What  must  be  the  diameter  of  a  single  pipe  to  carry  away  all  of 
the  water  ? 

195.  A  and  B  bought  a  ball  of  twine,  8  inches  in  diameter,  for 
$  I ;  A  wound  from  the  outside  until  the  diameter  of  the  part 
that  was  left  was  4  inches.     How  much  should  each  pay  ? 


272 


PROOFS  AND  ILLUSTRATIONS 


PROOFS  AND  ILLUSTRATIONS  —  LINEAR  PARTS 

I.  The  circumference  of  a  circle  is  equal  to  ttcice  the  radius 
times  3. 1 416. 

For  proof,  the  pupil  is  referred  to  geometry.  We  may  illustrate  by 
measurement  Take  a  circle,  as  the  base  of  a  pail ;  with  string  and  rule, 
measure  its  circumference ;  measure  its  diameter ;  divide  the  circumference 
by  the  diameter;  the  quotient  will  be  3.1410  approximately. 

II.  The  square  of  the  ht/potenuse  of  a  right-angled  tnangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides. 


For  proof,  the  pupil  is  referred  to  geometry.  We  may  illustrate  by 
measurement.  Draw  two  lines  at  right  angles ;  from  the  right  angle,  lay 
off  .3  in.  on  one  line,  4  in.  on  the  other,  and  connect  the  extremities ;  by 
measurement,  the  hypotenuse  will  be  6  in. ;  5*  =  3^  +  4^. 


AREAS 

III.    The  area  of  a  rectangle  is  equal  to  the  product  of  its  base 
by  its  altitude. 


This  rectangle  may  be  regarded  as  separated  into  strips  by  lines  parallel 
to  the  base.  The  number  of  squares  in  each  layer  is  the  same  as  the  number 
of  linear  units  in  the  base  ;  the  number  of  squares  in  each  strip  multiplied 
by  the  number  of  strips,  is  the  number  of  squares  in  the  rectangle ;  .-.  the 
area  of  a  rectangle  is  equal  to  the  product  of  its  base  and  its  altitude. 


MENSURATION 


273 


IV.    The  area  of  a  parallelogram  is  equal  to  tlie  product  of  its 
base  by  its  altitude. 

B C 


Let  ABCD  be  a  parallelogram,  and  BE  its  altitude. 

To  prove  that  its  area  =  AD  x  BE. 

Draw  CF  parallel  to  BE  and  prolong  AD  X<i  F.  The  right  triangles  ABE 
and  DCF  are  equal,  because  AB  =  DC,  BE  =  CF,  and  AE  =  DF.  From 
the  whole  figure,  take  away  the  triangle  ABE ;  the  rectangle  EBCF  re- 
mains. From  the  whole  figure,  take  away  the  triangle  CDF ;  the  parallelo- 
gram remains.  Therefore,  the  parallelogram  has  the  same  area  as  the 
rectangle.  The  area  of  the  rectangle  =  BCx  BE ;  .*.  the  area  of  the  parallelo- 
gram =  AD  X  BE. 

V.  The  area  of  a  triangle  is  equal  to  one  half  the  product  of  its 
ba^e  by  its  altitude. 


Let  ^BC  be  a  triangle,  and  BD  its  altitude. 

To  prove  that  its  area  =  i  AC  x  BD. 

Draw  BE  parallel  to  AC,  and  E.i  parallel  to  BC.  EBCA  is  a  parallelo- 
gram. Triangles  ABE  and  ABC  are  equal,  because  AB  =  AB,  EA  =  BCj 
and  EB  =  AC.  Therefore,  triangle  ABC  is  one  half  of  the  parallelogram. 
The  area  of  the  parallelogram  is  ^C  x  BD,  therefore,  the  area  of  the  triangle 
ABC=\ACx  BD. 

VI.  The  area  of  a  triangle  is  equal  to  the  square  root  of  the  con- 
tinued product  of  the  half  sum  of  its  sides  and  the  remainders  found 
by  8\ibtracting  each  side  from  the  half  sum  separately. 

For  proof,  the  pupil  is  referred  to  geometry.  We  may  illustrate  by  find- 
ing the  area  of  a  right-angled  triangle  by  each  rule.  The  sides  of  a  right- 
angled  triangle  are  3,  4,  and  6.    By  the  first  rule,  the  area  is  ^  x  3  x  4,  or  6 ; 

3-t-4-f  5 
2 


by  the  second,  the  half  sum  is 


:  6 ;  remainders  are  6  -  3,  6  —  4, 


6  -  6,  or  3,  2,  1 ;  area  is  V6  x  3  x  2  x  1,  or 

▲MSB.    ARITH.  —  18 


274 


PROOFS   AND   ILLUSTRATIONS 


AREAS 

VII.    Tfie  area  of  a  trapezoid  is  equal  to  one  half  the  product  of 
the  sum  of  its  jnirallel  sides  by  its  altitude. 

A  B 


ED  C 

SDOOE8TION.     Area  AEC  -  \  AD  x.  EC  -,  area  ABC  =\AD  y.  AB. 

VI IL    The  area  of  a  regular  polygon  is  equal  to  one  half  the 
product  of  its  jyerimeter  by  its  apothem. 

IX.    The  area  of  a  circle  is  equal  to  the  square  of  its  radius  times 
3.1U6, 


Let  ABCDEF  be  a  reg;ular  polygon,  and  OM  its  apothem. 

To  prove  that  its  area  =  J  ABCDEF  x  OM. 

From  the  center  of  the  regular  polygon  draw  lines  to  the  vertices.  ITiey 
divide  the  regular  polygon  into  equal  triangles.  The  area  of  the  regular 
polygon  is  the  sura  of  the  areas  of  the  triangles.  The  area  of  a  triangle  =  I 
the  product  of  its  base  by  its  altitude  ;  the  sum  of  the  bases  of  the  triangles 
is  the  perimeter  of  the  polygon  ;  the  altitude  {OM)  of  one  of  the  triangles  is 
the  apothem  of  the  polygon.  Therefore,  the  area  of  the  regular  polygon  is 
equal  to  \  ABCDEF  x  OM. 

The  area  of  the  circle =^abcdefx om ;  abcdef=2  xRx 3.1416(L) ;  om=R ; 
.-.  areaof  the  circle  =  ix2xJ?x3. 1416 xi?=i?2x 3. 1416. 

X.  The  area  of  an  irregular  polygon  is  equal  to  the  sum  of  the 
areas  of  the  triangles  into  which  it  may  be  divided. 

The  proof  is  self-evident.    The  pupil  should  draw  an  irregular  polygon. 


MENSURATION 


275 


CONVEX   SURFACES 

XI.  The  convex  surface  of  a  prism  is  the  product  of  the  perimeter 
of  its  base  by  its  altitude. 

XII.  The  convex  surface  of  a  cylinder  is  the  product  of  the  cir- 
cumference of  its  base  by  its  altitude. 


\A 


M/ 


The  faces  of  this  prism  are  rectangles ;  the  sum  of  the  rectangles  is  the 
convex  surface  of  the  prism  ;  the  area  of  a  rectangle  is  the  product  of  its  base 
by  its  altitude  ;  .  •.  the  convex  surface  of  a  prism  is  the  product  of  the  perimeter 
of  its  base  by  its  altitude. 

The  cylinder  is  a  variety  of  the  prism  ;  .  •.  the  convex  surface  of  a  cylinder 
is  the  product  of  the  circumference  of  its  base  by  its  altitude. 

XIII.  The  convex  surface  of  a  pyramid  is  half  the  product  of  the 
perimeter  of  its  base  by  its  slant  height. 

XIV.  Tfie  convex  surface  of  a  cone  is  half  the  product  of  the  cir- 
cumference of  its  base  by  its  slant  height. 


The  faces  of  this  pyramid  are  triangles ;  the  sum  of  the  triangles  is  the 
convex  surface  of  the  pyramid  ;  the  area  of  a  triangle  is  half  the  product  of 
its  base  by  its  altitude ;  the  altitude  of  one  of  the  triangles  is  the  slant  height 
of  the  pyramid  ;  .*.  the  convex  surface  of  a  pyramid  is  one  half  the  product 
of  the  perimeter  of  its  base  by  its  slant  height. 

The  cone  is  a  variety  of  the  pyramid  ;  .  •.  the  convex  surface  of  a  cone  is 
one  half  the  product  of  the  circumference  of  its  base  by  its  slant  height. 


276  PROOFS   AXD  ILLUSTRATIONS 

XV.  The  convex  surface  of  a  frustum  of  a  pyramid  is  half 
the  product  of  the  sum  of  the  perimeters  of  its  two  bases  by  its 
slatit  heiglU. 

XVL  The  convex  surface  of  a  frustum  of  a  cone  is  half  the  prod- 
uct of  the  sum  of  the  circumferences  of  its  two  bases  by  its  slant  heigfU. 


The  faces  of  this  frustum  are  trapezoids ;  the  sum  of  the  trapezoids  is  the 
convex  surface  of  the  frustum  ;  the  area  of  a  trapezoid  is  the  product  of  half 
the  sum  of  its  parallel  sides  and  its  altitude  ;  the  altitude  of  one  of  the 
trapezoids  is  the  slant  height  of  the  pyramid  ;  therefore,  the  convex  surface 
of  the  frustum  of  a  pyramid  is  half  the  product  of  the  sum  of  the  perimeters 
of  its  two  bases  by  its  slant  height. 

The  frustum  of  a  cone  is  a  variety  of  the  frustum  of  a  pyramid  ;  therefore, 
the  convex  surface  of  the  frustum  of  a  cone  is  half  the  product  of  the  sum  of 
the  circumferences  of  its  two  bases  by  its  slant  height. 

XVII.    The  surface  of  a  sphere  is  four  times  the  square  of  its 

radius  times  S.Ij^IS. 

The  proof  is  too  difficult  for  introduction  here.    See  any  good  geometry. 

VOLUMES 

XVI II.,  XIX.  The  volume  of  a  prism  or  cylinder  is  equal  to  the 
product  of  the  area  of  its  base  by  its  altitude. 


Q 


This  prism  may  be  regarded  as  separated  into  layers  by  planes  parallel  to 
the  base.  The  number  of  cubes  in  each  layer  is  the  same  as  the  number  of 
squares  on  the  surface  of  the  base ;  the  number  of  cubes  in  each  layer  mul- 
tiplied by  the  number  of  layers  is  the  number  of  cubes  in  the  prism  ;  there- 
fore, the  volume  of  a  prism  is  equal  to  the  product  of  the  area  of  its  base  by 
its  altitude.  The  cylinder  is  a  variety  of  the  prism  ;  therefore,  the  volume 
of  a  cylinder  is  equal  to  the  product  of  the  area  of  its  base  by  its  altitude. 


MENSURATION 


277 


XX.,  XXI.     The  volume  of  a  pyramid  or  cone  is  equal  to  one 
third  the  product  of  the  area  of  its  ba^e  by  its  altitude. 


The  proof  is  too  difficult  for  introduction  here,  but  the  rule  may  be  veri- 
fied by  making  a  hollow  pyramid,  and  a  hollow  prism  with  equal  bases  and 
altitudes.  Ii  will  be  found  that  the  pyramid  will  hold  one  third  as  much  as 
the  prism. 

The  cone  is  a  variety  of  the  pyramid  ;  therefore,  the  volume  of  a  cone  is 
equal  to  one  third  the  product  of  the  area  of  its  base  by  its  altitude. 

XXII.  The  volume  of  a  frustum  of  a  pyramid  is  equal  to  one 
third  the  product  of  the  sum  of  the  areas  of  its  upper  ba^se^  lower 
base,  and  mean  proportional  base,  by  its  altitude. 

XXIII.  The  volume  of  a  frustum  of  a  cone  is  equal  to  one  thii-d 
the  product  of  the  »um  of  the  areas  of  its  upper  Ixise,  lower  base,  and 
mean  proportional  base,  by  its  altitude. 


The  proof  is  too  difficult  for  introduction  here,  but  the  rule  may  be  veri- 
fied by  making  a  hollow  frustum  of  a  pyramid,  and  three  hollow  pyramids ; 
one  having  the  lower  base  of  the  frustum,  one  the  upper  base,  and  one  the 
mean  proportional  base,  and  all  having  the  same  altitude.  It  will  be  found 
that  the  frustum  will  hold  as  much  as  the  three  pyramids  together. 

The  frustum  of  a  cone  is  a  variety  of  the  frustum  of  a  pyramid  ;  therefore, 
the  volume  of  a  frustum  of  a  cone  is  equal  to  one  third  the  product  of  the 
sum  of  the  areas  of  its  upper  base,  lower  base,  and  mean  proportional  base 
by  its  altitude. 


278  PROOFS   AND  ILLUSTRATIONS 

VOLUMES 

XXIV.     Tlie  volume  of  a  sphere  is  equal  to  four  thirds  times  the 
cube  of  its  radius  times  S.14^6. 


A  sphere  may  be  regarded  as  made  up  of  equal  pyramids.  The  sura  of 
the  bases  of  the  pyramids  is  the  surface  of  the  sphere  ;  the  altitude  of  each 
pyramid  is  the  radius  of  the  sphere  ;  the  sum  of  the  volumes  of  the  pyramids 
is  the  volume  of  the  sphere.     Therefore, 

The  volume  of  a  sphere  =  surface  x  \R. 

(Since  surikce  -4  X  R*  x S.Ulft.) 

Volume  =  4  X  i?2  X  3.1416  x  ^  x  5,  or  j  x  i?«  x  3.1416. 


SIMILARITY 

XXV.-XXVII.  The  laws  of  similarity  are  deduced  from  many 
propositions  in  geometry. 

DIMENSIONS 

XXVIII.     There  caii  be  only  three  dimensions. 

By  this  is  meant,  that  only  three  lines  can  be  drawn  at  a  given  point, 
each  of  which  is  perpendicular  to  each  of  the  others.  This  truth  is  self- 
evidenL 


OCCUPATIONS 


WITH   THE   LUMBER   DEALER 

The  classification  of  lumber  is  not  exact,  but  the  following  may 
prove  helpful: 

Lumber  —  wooden  building  material. 

I.   Boards — 1  in.  thick  —  less  if  specified. 

1.  St(x;k  boards ;  boards  of  uniform  width,  12  in.  wide. 

2.  Fencing ;  6  in.  wide. 

3.  Flooring ;  matched  boards. 

4.  Siding  or  clapboards;    J  in.  thick,  thicker  at  one 

edge. 

n.    Dimension  Stuff —  more  than  1  in.  thick. 

1.  Scantling;  2  in.  to  4  in.  thick,  3  in.  to  4  in.  wide. 

2.  Joist;  2  in.  thick,  any  width. 

3.  Plank ;  2  in.  thick,  wider  than  4  in. 

4.  Timber ;  thicker  than  2  in.,  wider  than  4  in. 

III.  Foot  Stuff— -sold  by  linear  foot. 

1.  Battens  for  covering  cracks. 

2.  Molding  for  finishing. 

IV.  Laths  — 4  ft.  long,  IJ  in.  wide,  50  to  a  bundle. 

V.  Shingles  — i  in.  wide,  250  to  the  bunch.  They  are  not  of 
uniform  width,  but  every  4  in.  is  reckoned  as  one 
shingle. 

879 


280 


WITH   THE   LUMBKR  DEALER 


Lumber  is  sold  by  the  board  foot 
A  board  foot  is  the  equivalent  of  1 
ft.  long,  1  ft.  wide,  and  1  in.  thick. 

Inch  lumber  12  ft.  long  contains  as 
many  board  feet  as  there  are  inches  in 
its  width. 


1  board  ft. 

Because    12    ft.  x  ^^    ft. 
X  1  in.  =  1  ft.  X  1  ft.  X  1  in. 


Uoiv  mxiny  feet  of  lumber  are  there  in  : 


1.  1  8  in.,  12  ft  board?  U. 

2.  1  8  in.,  10  ft  board  ?  12. 

3.  1  8  in.,  6  ft.  board  ?  la 

4.  1  8  in.,  14  ft  board  ?  14. 

5.  1  8  in.,  18  ft  board  ?  15. 
R  3  6  in.,  10  ft.  boards  ?  1& 
7.  4  7  in.,  16  ft  boards  ?  17. 
a  1  joist  2  X  4,  12  ?  la 
9.  1  joist  2  x4, 16?  19. 

10.  1  scantling  3  x  4, 12  ?  20. 


1  plank  2x10, 12? 
1  plank  2  x  10, 16  ? 

4  timbers  4  x  4,  20  ? 
3  pieces  8  x  8,  24  ? 

5  pieces  2  x  4, 12  ? 

8  3  in.,  12  ft.  boards  ? 

6  14  ft  fencing  ? 

6  pieces  4  x  6,  20  ? 
10  6  in.,  12  ft  siding? 
10  scantlings  3  x  4,  16 


Ex.  2.  7.  An  8  in.  12  ft.  board  would  contain  8  ft. ;  10  =  12  -  |  of  12  ; 
8  —  ^of8  =  6j;  taking  the  nearest  whole  number,  7. 

Ex.  7.  37.   4  7  in.  boards  =  1  28  in.  board.     A  28  in.  12  ft.  board  would" 
contain  28  ft. ;  16  =  12  +  ^  of  12  ;  28  +  ^  of  28  =  37^ ;  taking  the  nearest 
whole  number,  37. 

Ex.  9.  11.  This  is  read,  "  1  joist  2  by  4,  IC."  The  joist  is  2  in.  thick, 
4  in.  wide,  16  ft.  long. 

Ex.  13.  107.  4  4x4  pieces  =  1  64  in.  board.  A  64  in.  12  ft.  board 
would  contain  64  ft.  ;  20  =  12  +  f  of  12 ;  64  +  f  of  64  =  106| ;  taking  the 
nearest  whole  number,  107. 


Note. — Unless  otherwise  specified,  stuff  less  than  an  inch  thick  is  comited  as 
an  inch  thick.    See  Ex.  19. 


OCCUPATIONS 


281 


Dealers  sometimes  use  a  card  like  the  following,  carried  out 
for  a  great  variety  of  lengths  and  dimensions.  Usually,  as  in 
this  table,  fractions  of  a  foot  are  neglected. 


Lumber  Table 


SI/E 

12  14  16  18  1 20  22 1  24  26  28  30  32  34  36  38  1 40 

8x6 

12 

14 

16 

18 

20 

22 

24 

20 

28 

30 

32 

34 

36 

38 

40 

2x8 

16 

19 

21 

24 

27 

20 

32 

35 

37 

40 

43 

45 

48 

51 

53 

8x4 

4x6 

8x4 

8x3 

21.  Verify  the  results  in  the  1st  horizontal  line. 

22.  Verify  the  results  in  the  2d  horizontal  line. 

23.  Declare  the  results  for  the  3d  horizontal  line. 

24.  Declare  the  results  for  the  4th  horizontal  line. 

25.  Declare  the  results  for  the  5th  horizontal  line. 

26.  Declare  the  results  for  the  6th  horizontal  line. 


How  many  hoard  feet  are  there  in : 
27.  4plk.,  2  x4,  12? 
2a  2  timbers,  4  x  6,  40  ? 
29.  9  joists,  2  x3,  16? 
aa  12  bds.,  1  X  12, 16? 

31.  24  fencing,  1  x  6,  16? 

32.  4  piece  stuff,  3  x3,  12? 

33.  6  scantlings,  2  x  4,  16? 


34.  3  timbers,  6  x  6,  40? 

35.  2  timbers,  8  x  8,  60? 

36.  3  joists,  2  x8,  12? 

37.  6plk.,  Hxl2,  12? 

38.  6plk.,  If  X  12,12? 

39.  6plk.,  I}xl2,  12? 

40.  6  plk.,  1 J  X  12,  12  ? 


Note.  ~  Count  lumber  less  than  1  in.  thick  as  1  in. ;  from  11  to  \\,  as  \\ ;  from 
11  to  II,  as  11;  from  11  to  2.  as  2. 


282  WITH   THE   LUMBER  DEALER 

Boards,  Planks,  Scantliko 

41.  To  build  120  ft.  of  sidewalk,  4  ft.  wide,  I  use  2x4  scantlings, 
(3  rows  running  lengthwise).  How  many  feet  of  scantling  will 
be  required  ?     How  jnany  feet  of  boards  ? 

42.  Which  is  the  best  length  of  boards  in  Ex.  41,  14,  or  16  ft.? 
Which  is  the  best  length  of  scantling,  12,  14,  or  IG  ft.  ? 

4a  How  many  feet  of  stock  boards  (12  in.  wide)  will  be  re- 
quired to  cover  a  barn  30  ft.  by  40  ft,  16  ft.  posts,  quarter-pitch 
X  roof,  making  no  allowance  for  doors  or  windows  ? 


_  NoTK.  —  A  roof  has  a  quarter-pitch  when  the  height  of  the 

B        t)        C    ^,j^j,j^.^  ^jj  |g  J  of  the  width  of  the  building,  BC. 

44.  How  many  feet  of  boards,  not  counting  waste,  will  cover 

both  gables  of  a  building  30  ft.  wide,  and  having  a  quarter-pitch 

roof? 

Battens 

45.  How  many  linear  feet  of  battens,  not  counting  the  gables, 

will  be  required  to  cover  the  cracks  in  the  barn  of  Ex.  43  ? 

Note.  —  Battens  at  the  comers  are  not  necessary,  bat  are  generally  used  for 
appearance. 

46.  At  $  10  per  M,  what  is  the  cost  of  the  boards  in  Ex.  43  ? 

Siding 

47.  If  6  in.  clapboards  are  laid  4  in.  to  the  weather,  how  many 
feet  of  clapboards,  no  allowance  for  waste,  will  be  required  to 
cover  1000  sq.  ft.  of  surface  ? 

4a  How  does  your  result  agree  with  the  rule,  "  To  the  sq.  ft.  in 
the  surface  add  ^  the  surface  "  ? 

Matched  Flooring 

49.  How  many  feet  of  16  ft.  6  in.  matched  flooring,  no  allow- 
ance for  waste,  will  be  required  for  a  room  16  ft.  square  ? 

Note.  —  The  width  of  one  piece  of  flooring  is  C  in.,  but  since  this  includes  the 
tongue,  one  piece  covers  about  5  in.  only. 

50.  How  does  your  result  agree  with  the  rule,  "  To  the  sq.  ft.  in 
the  surface,  add  |  the  surface  ^^  ? 


OCCUPATIONS  283 

Laths 

51.  If  laths  are  laid  f  of  an  in.  apart,  how  many  square  inches 
(including  the  space  between  laths)  does  1  lath  cover  ? 

52.  How  many  square  yards,  no  allowance  for  waste,  will  1 
bundle  of  lath  cover  ? 

53.  How  does  this  agree  with  the  rule,  *'  One  bundle  of  lath  will 
cover  S  sq.  yd."  f 

54.  How  many  bundles  of  lath  will  be  required  for  the  ceiling 
and  sides  of  a  room  18  x  20  x  8  ft.,  no  allowance  for  openings  ? 

Shingles 

55.  If  shingles  are  laid  4  in.  to  the  weather,  no  allowance  for 
waste,  how  many  square  feet  will  1  bunch  of  shingles  cover  ? 

56.  How  does  this  agree  with  the  rule,  "  4  bunches  of  shingles 
laid  4  »»•  lo  the  weather  will  cover  100  sq.  Jl."  ? 

57.  If  shingles  are  laid  4J^  in.  to  the  weather,  no  allowance  for 
waste,  how  many  square  feet  will  1  bunch  cover  ? 

5a  How  does  your  result  agree  with  the  rule,  "900  shingles 
laid  4\  ii^-  lo  the  weather  will  cover  100  sq.Ji."? 

59.  How  many  bunches  of  shingles,  laid  4  in.  to  the  weather, 
will  be  required  for  the  barn  mentioned  in  Ex.  43,  supposing  the 
rafters  to  project  1  ft.,  and  the  roof  to  project  16  in.  at  each  end  ? 

Miscellaneous 

ea  How  many  linear  feet  of  battens  are  required  for  a  barn 
40  ft.  X  40  ft.,  the  posts  22  ft.,  the  pitch  a  quarter,  the  boards 
being  12  in.  wide  ?     Batten  the  gables. 

61.  How  many  board  feet  are  there  in  40  pieces  16  ft.  siding  ? 

62.  How  many  feet  of  6  in.  matched  floorintr  will  be  required 
for  a  room  16  x  20  ft.  ? 

63.  If  lathing  is  laid  |  of  an  inch  apart,  how  many  square 
yards  will  4  bundles  of  lath  cover  ? 

64.  If  shingles  are  laid  4  in.  to  the  weather,  how  many  square 
feet  will  4  bunches  cover  ? 


284 


WITH   THE   LUMBER  DEALER 


Measurement  of  Loos 

By  the  number  of 
board  feet  in  a  log, 
is  meant  the  number 
of  board  feet  in  the 
largest  piece  of  tim- 
ber that  can  be  sawed 
from  the  log. 

A  piece  1  in.  x  1  in.  x  12  ft 
(usually  written  1x1,  12)  contains 
1  board  foot. 


A  log  12  ft.  long  contains  as  many 
hoard  feet  as  there  are  sq.  inches  on  its 
squared  end,  or  as  many  board  feet  as 
there  are  sq.  inches  in  half  the  square 
of  its  diameter. 

If  the  log  is  to  he  sawed  into  hoards, 
dedu^ct  J  for  waste. 


4  board  feet. 


The  area  of  the  greatest  in- 
scribed square  is  AB^. 

Alf-{-A^  =  Bd^ 
(sq.  of  hyp.  =  sum  ofsqs.) 
tAi^  =  BC^ 

2 


AB'  = 


How  many  feet  of  lumber  are  there  in  a  log : 


65.  Length  12  ft.,  D.    8  in.  ? 

66.  Length  16  ft.,  D.  10  in.  ? 

67.  Length  20  ft.,  D.  12  in.  ? 


6a   Length  24  ft.,  D.  10  in.  ? 

69.  Length  32  ft.,  D.  18  in.  ? 

70.  Length  16  ft.,  D.  12  in.  ? 


Ex.  66.  67  ft.  of  lumber.     10^  =  100  ;  ^  of  100  =  60  ;  if  the  log  were  12  ft. 
long,  there  would  be  50  board  ft. ;  60  +  J  of  50  =  67. 


How  Tnany  feet  of  boards  are  there  in  a  log  : 

71.  Length  12  ft.,  D.    8  in.  ?         74.   Length  20  ft.,  D.    6  in.  ? 

72.  Length  18  ft.,  D.  10  in.  ?         75.   Length  36  ft.,  D.  10  in.  ? 

73.  Length  24  ft,  D.  12  in.  ?         76.   Length  40  ft,  D.  16  in.  ? 

Ex.  71.  20  ft.    There  would  be  32  ft.  of  lumber ;  32  -  ^of  32  =  26. 


OCCUPATIONS  285 

WITH  THE  CARPET  DEALER 

Carpeting  is  made  of  various  widths,  and  sold  by  the  linear 
yard. 

To  determine  the  number  of  yards  of  carpet  for  a  roomy  decide 
whether  the  breadths  shall  rmi  lengthwise  or  crosswise,  find  the  num- 
ber ofbreadtlis  required  and  multiply  by  the  length  of  a  breadth. 

NoTK.  —  Allowance  should  also  be  made  for  waste  iu  matching. 

77.  How  many  yards  of  carpet  J  of  a  yard  wide,  will  be  required 
for  a  room  16  ft.  long  and  13  ft.  wide,  the  breadths  running 
lengthwise,  no  waste  in  matching  ? 

Ans.  32  yd.  }  yd.  =  2\  ft.  ;  13  ft.  -h  2J  ft.  =  5J ;  .-.  6  strips  will  be  re- 
quired ;  16  ft.  X  6  =  96  ft. ,  or  32  yd. 

7a   In  Ex.  77,  how  many  square  yards  must  be  turned  under  ? 

79.  A  floor  18  ft.  by  16  ft.  is  to  be  covered  with  carpeting  J  yd. 
wide.  Which  will  be  the  more  economical,  to  run  the  breadths 
lengthwise  or  crosswise,  if  there  is  a  waste  in  matching  of  6  in. 
on  each  breadth  except  the  first  ?  Why  is  there  no  waste  on  the 
first  breadth  ? 

80.  At  75^  per  yard,  how  much  will  it  cost  to  carpet  a  room 
19  ft.  by  15  ft.  with  carpeting  }  yd.  wide,  the  same  figure  recur- 
ring at  intervals  of  8  in.,  and  the  breadths  running  lengthwise  ? 

81.  You  can  find  carpeting  of  the  same  grade  which  pleases 
you  equally  well,  f  yd.  wide  at  75^  per  yd.,  or  1  yd.  wide  at  $  1 
per  yard,  no  waste  in  matching  in  either  case.  Your  room  is 
16  ft.  by  14  ft. ;  which  way  should  the  breadths  run,  and  which 
width  of  carpet  ought  you  to  buy  to  carpet  the  room  with  least 
expense  ?    What  would  be  the  least  expense  ? 

82.  At  95^  a  yard,  what  will  be  the  cost  of  carpeting  a  room 
IS  X  20  ft.  with  carpet  1  yd.  wide,  if  the  breadths  run  lengthwise, 
and  there  is  a  loss  in  matching  of  8  in.  to  a  breadth  ? 

8a  What  will  be  the  cost  of  carpeting  the  room  mentioned  in 
Ex.  82,  with  I^rus.sels  carpet  }  yd.  wide  at  $  1.26  ?  The  breadths 
are  to  run  crosswise,  and  there  is  to  be  no  waste  in  matching. 


286  WITH    THK    PAPER  HANGER 

WITH  THE   PAPER  HANGER 

Wall  paper  is  sold  by  the  double  roll)  48  ft.  x  IJ  ft.,  or  by  the 
single  roU,  24  ft.  xl\  ft.  It  is  matched,  cut  up  into  strips,  and 
pasted  upon  the  walls  or  ceiling. 

From  the  distance  around  the  room  in  feety  deduct  3  ft.  for  each 
opening  (door  or  window).  The  remainder  -i- 1  will  give  the  num- 
ber of  strips  required  for  the  walls. 

The  walls  and  ceiling  of  an  8  foot  room,  20  ft  x  16  ft.,  are  to 
be  papered ;  there  are  four  windows  and  a  door. 

84.  How  many  strips  for  the  walls  will  a  double  roll  make? 
Explain. 

Ana.  6  strips.  There  is  a  loss  in  matching,  but  this  need  not  be  con- 
sidered, because  the  paper  docs  not  extend  to  the  floor,  on  account  of  the 
base  board,  nor  to  tlie  ceiling,  on  account  of  the  border. 

85.  By  the  rule,  how  many  double  rolls  will  be  needed  for  the 
walls  ?     Explain. 

86.  If  the  strips  run  lengthwise,  how  many  strips  for  the  ceil- 
ing will  a  double  roll  make  ?     Explain. 

87.  If  the  strips  run  crosswise,  how  many  strips  for  the  ceil- 
ing will  a  double  roll  make  ?     Explain. 

8a  How  many  double  rolls  will  be  required  for  the  ceiling, 
if  the  strips  run  lengthwise  ? 

89.  How  many  double  rolls  will  be  required  for  the  ceiling, 
if  the  strips  run  crosswise  ? 

90.  What  will  be  the  cost  of  papering  this  room,  walls  and 
ceiling,  at  36^  a  double  roll,  with  border  around  the  walls  at  3^ 
per  linear  foot  ? 

91.  Do  we  ever  in  practice  find  the  exact  number  of  square 
feet  on  the  walls,  deduct  for  the  doors  and  windows,  and  then 
divide  by  the  number  of  square  feet  in  a  double  roll  ?  Why 
not? 


OCCUPATIONS  287 

WITH  THE  MASON 

Bricks  are  usually  8  in.  x  4  in.  x  2  in. 

In  estimating  amount  of  work,  masons  measure  the  length  of  walls 
on  the  outside,  thus  counting  the  coiners  twice.  They  consider 
this  just  on  account  of  the  greater  labor  of  construction.  Except 
by  special  contract,  no  allowance  is  made  for  windows  and  doors. 

In  estimating  amount  of  material,  the  corners  are  measured 
twice,  but  allowance  for  windows  and  doors  must  be  made. 

92.  How  many  bricks,  increasing  the  length  and  thickness  J  in. 
for  mortar,  will  lay  1  cubic  foot  ? 

Ans.  2^^j.  There  are  8^  x  4  x  2}  cu.  in.  in  1  brick ;  1728  x  ,*j  x  }  x 
I  =  23i'r,  the  no.  of  bricks. 

NoTK.  —  Experience  shows  that  22  bricks  8x4x2  will  lay  1  cu.  ft. 

93.  How  many  bricks  will  be  required  to  build  the  walls  of  a 
house  60  X  40  X  34  ft.,  outside  measure,  2  bricks  thick.  There 
are  20  windows  6  x  3  f t. ;  and  6  doors,  6x3^  ft. 

goo  =ft.  around  house  20  x  6  y,  S  x\  =  S40  cu,  ft,  windows^ 

g00xS4x\=4533{=cu.  ft.  walls  ^     ^1     s        „.        ^   ^ 

*       -^/'          ^                       6x6xSix^=    84  cu.  ft.  doors 
S24  =  cu.  ft.  openings  «     s      

4209^-  =  cu.  ft.  clear  ^^*  ^-  ■^-  «P«'»»»f'» 

gg^  NoTK.  — Wall  is  2  bricks,  or  }  ft. 

thick. 


92605  =  no.  of  bricks 

94.  For  how  many  cubic  feet  would  a  mason  charge  for  build- 
ing the  walls  in  Ex.  93? 

Ans.   4533]|.     lie  would  not  allow  for  openings. 

95.  If  the  mason  laid  the  brick  per  M  (by  the  thousand),  for 
how  many  thousand  would  he  charge  ? 

Ans.  90.733  thousand.    4633J  x  22  h-  1000  =  90.733.    He  would  not  allow 
for  openings. 

96.  How  much  sand  and  lime  will  be  required  for  the  building  ? 
Note.  — U  bbl.  of  lime  and  |  cu.  yd.  of  sand  will  lay  1000  bricks. 


288  WITH   THE   MASON 

97.  How  many  masons  will  be  required  to  build  the  walls  of 
the  house  in  Ex.  93,  in  11  days? 

NoTB.  —  One  mason  will  lay  about  1800  bricks  per  day. 

9a  How  many  bricks  will  be  required  to  build  a  house  25  x  80, 
36  ft.  high,  if  the  walls  are  3  bricks  thick,  and  400  cu.  ft.  is  to  be 
allowed  for  openings  ? 

99.   How  much  will  it  cost  to  lay  the  brick  at  $  4.50  per  M  ? 

100.  How  many  loads  of  sand  will  be  required?     How  many 
barrels  of  lime  ?    In  how  many  days  will  six  masons  do  the  work  ? 

101.  If  there  are  two  chimneys  50  ft.  high  and  2  ft.  square,  out- 
side measure,  how  many  bricks  will  be  required  for  the  chimneys  ? 

NoTK.  —  Consider  each  chimney  solid. 

102.  How  many  bricks  are  required  for  the  4  in.  walls  of  a  cistern 
12  X  8  X  8  ft.  ? 

103.  How  many  bricks  are  required  for  the  4  in.  walls  oi  a  cistern 
24  X  16  X  16  ft.  ? 

104.  In  Ex.  103  why  is  the  answer  not  twice  as  many  bricks 
as  in  Ex.  102  ? 

105.  How  many  cords  of  stone  are  required  for  the  foundation 
of  the  house  in  Ex.  98 ;  the  walls  to  be  10  ft.  high  and  2  ft.  wide  ? 

Note.  —  A  cord  of  stone  (128  cu.  ft.)  will  lay  100  cu.  ft. 

106.  For  how  many  perch  of  stone  would  the  mason  charge  in 
building  the  above  foundation  ? 

Note.— 24i  cu.  ft.  make  one  perch. 

107.  How  much  sand  and  lime  will  be  required  for  the  founda- 
tion? 

NoTS.  — 1\  bbl.  of  lime  and  1  cu.  yd.  of  sand  will  lay  100  cu.  ft.  of  stone. 
loa   What  would  be  the  cost  of  plastering  a  room  40  x  30,  12 
ft.  high,  at  30^  a  sq.  yd. ;  8  windows  3  x  6,  2  doors,  6x4? 
Note.  —  Do  not  deduct  for  openings. 

109.   How  much  lime  and  sand  would  be  required  for  two  coats  ? 
Note.  —  34  bbl.  of  lime  and  li  cu.  yd.  of  sand  will  plaster  100  sq.  yd.,  2  coats. 


OCCUPATIONS  289 

WITH   THE   FARMER 
Bins 

110.  How  many  bushels  of  wheat  will  a  bin  12  x  8  x  6  ft. 
contain  ? 

Note.  —  GraiD  is  sold  by  struck  measure ;  4  struck  ba.  =  5  cu.  ft. 

111.  How  many  bushels  of  potatoes  will  a  bin  12  x  8  x  6  ft. 
contain  ? 

Note.  —  Potatoes  are  sold  by  heaped  measure ;  4  heaped  bu.  =  5  struck  bu. 

112.  How  many  bushels  of  shelled  corn  will  a  bin  12  x  8  x  6  ft. 
contain  ?     How  much  will  it  weigh  ? 

Note.  —  Shelled  com  is  sold  by  the  struck  bushel ;  56  lb.  =  1  bu. 

113.  How  many  bushels  of  shelled  corn  may  be  obtained  from 

a  bin  12  x  18  x  6  ft.,  tilled  with  coin  in  the  ear? 

Note.  —  3800  to  4000  cu.  in.  corn  in  ear  make  1  bushel  shelled  corn.  For  small 
bins,  7  cu.  ft.  =  3  bu.  shelled  corn. 

114.  How  many  bushels  of  oats  are  there  in  a  conical  pile, 
altitude,  9  ft. ;  circumference  of  the  base,  20  ft.  ? 

115.  A  crib  whose  lower  base  is  8  x  10  ft.,  and  whose  upper 
base  is  12  X  15  ft.,  depth,  12  ft.,  is  filled  with  corn  in  the  ear. 
How  many  bushels  of  shelled  com  may  be  obtained  from  the 
crib  ? 

116.  How  many  bushels  of  shelled  com  may  be  obtained  from 
2100  lb.  of  old  corn  in  the  ear  ?    Of  new  ? 

Note.  — Count  70  lb.  of  old  corn,  and  75  lb.  of  new,  to  the  bushel. 

117.  How  many  bushels  of  potatoes  are  there  in  a  conical  ]^ile, 
altitude,  8  ft. ;  diameter  of  the  base,  8  ft.  ? 

lia  How  many  bushels  of  oats  are  there  in  a  bin  10^  6x4  ft, 
if  the  bin  is  J  full  ? 

119.  A  crib,  whose  lower  base  is  16x18  ft.,  upper  base  24x27 
ft.,  depth  12  ft.,  is  filled  with  corn  in  the  ear.  How  many  bins, 
12  X  10  X  6  ft.,  will  be  required  to  hold  the  com  after  it  is  shelled  ? 

▲MKB.    AHITH.  — 10 


290  VVITH   THE   FARMER 

Waoon  Boxes 

120.  A  wagon  box  is  10  ft.  x  3  ft.  How  many  bushels  cf 
wheat  will  it  hold  for  every  inch  in  depth  ? 

121.  How  many  bushels  of  rye  are  there  in  a  wagon  box  10  x  3 
ft.,  depth  of  rye,  12  in.? 

Ans.  24  bu.  rye.    In  Ex.  120,  2  bu.  were  found  for  each  inch  in  depth. 

122.  What  must  be  the  depth  of  a  wagon  box  10  x  3  ft.  to  hold 
50  bu.  of  shelled  corn  ? 

Cisterns 

123.  What  is  the  capacity,  in  barrels,  of  a  cistern  in  the  shape 
of  two  equal  frustums  of  cones  with  their  larger  bases  placed 
together ;  radius  of  smaller  base  3  ft,  radius  of  larger  base  6  ft., 
altitude  9  f t.  ? 

Hat 

124.  How  many  tons  of  hay  are  there  in  a  conical  stack,  the 
circumference  of  base  60  ft.,  and  distance  over  40  ft.,  counting 
(7)*  cu.  ft.  to  the  ton  ? 

.4 

A  Relations:    Circumference  =  2  x  CE  y.  3.1416  ;    AC^  = 

/  i  \  CE^  +  ^^- ;  vol.  =  ^  X  CE^  x  3.1416  x  AE. 

I    i    \  Solution.      CE  =  9.b;    CA  =  20;     AE=  17.6;     vol.  = 

/ — I — .A  1663.4  cu.  ft.                                Ans.  5  tons  approximately. 

125.  How  many  tons  of  hay  are  there  in  a  rick  40  by  20  ft., 
distance  over  44  ft.,  counting  400  cu.  ft.  to  the  ton  ? 

^B  =  40  ft.  Note.  —  Of  the  many  diflFer- 

BC  =  20  ft.  ent  rules  in  use,  the  Govern- 

Sj)(y  =  44  ft.  ment  Rule  is  the  best. 

GovKRNMENT  RcLB.  To  find  the  cu.  ft.  in  a  rick,  multiply  the  length,  by 
the  breadth,  by  half  the  difference  between  the  width  and  the  distance  over. 


OCCUPATIONS  291 

Plowing 

126.  With  a  16  iii.  plow,  not  counting  distance  in  turning,  how 
far  will  a  team  walk  in  plowing  1  acre  ? 

127.  With  a  14  in.  plow,  not  counting  distance  in  turning,  liuw 
fur  will  a  team  walk  in  plowing  1  acre '! 

Yield  per  Acre 

128.  If  corn  is  planted  in  rows  3  ft.  8  in.  apart,  each  way, 
how  many  hills  are  there  to  an  acre,  20  rd.  x  8  rd.  ? 

129.  If  each  hill  of  corn  yields  1  ear,  and  150  ears  make  a 
bushel,  what  is  the  yield  per  acre? 

130.  How  do  your  results  agree  with  the  rule,  "One  ear  of  com 
to  the  hill  yields  20  bu.  to  the  acre  "  ? 

Fencino 

131.  How  many  rods  of  fence  will  inclose  an  acre  in  the  form 
of  a  rectangle  1  rd.  x  160  rd.  ? 

132.  How  many  rods  of  fence  will  inclose  an  acre  in  the  form 
of  a  square  ? 

133.  How  many  rods  of  fence  will  inclose  an  acre  in  the  form 
of  a  circle  ? 

Miscellaneous 

134.  In  an  apple  orchard,  the  trees  are  planted  33  ft.  apart  each 
way.  How  many  trees  are  there  in  the  orchard,  if  it  is  20  rd. 
long,  and  contains  2  A.  ? 

135.  If  each  apple  tree  yields  6  bu.,  what  must  be  the  length 
of  a  bin  10  ft.  wide,  T)  ft.  deep,  to  contain  the  crop? 

136.  Approximately,  how  many  barrels  of  water  will  a  tank 
S  X  7  X  3  ft  contain  ?  In  what  time  will  it  be  filled  by  a  pipe 
which  discharges  10  gal.  au  hour  ? 


MISCELLANEOUS 


LONGITUDE  AND   TIMS 

I.  At  places  on  the  eartli's  surjlice  it  is  earlier  farther  west. 

When  the  sun  is  rising  at  a  place,  it  has  not  yet  risen  at  places  farther 
west.  Where  the  sun  has  not  yet  risen  it  is  earlier;  therefore,  it  is  earlier 
farther  west. 

II.  llie  sun  appears  to  pass  through  360°  in  24  hours. 
There  are  360°  in  the  earth's  circumference  and  24  hours  in  a  day. 

1.  The  longitude  of  Boston  is  71°  3' 30"  west,  and  that  of 
Chicago  87''  44'  30"  west     What  is  their  true  difference  in  time  ? 

Meaning.  Boston  is  71°  3'  30" 
west  of  the  meridian  of  Greenwich  ; 
Chicago,  87°  44'  30"  west. 

Solution.  Let  N8  represent  the 
meridian  of  Greenwich  ;  locate  Bos- 
ton and  Chicago  on  the  diagram. 
The  difference  in  longitude  between 
B  and  C  is  CD  -  BD,  or  16°  41'. 

Since  the  sun  appears  to  move 
through  360°  in  24  hr.,  16°  41'  =  1  hr. 
6  min.  44  sec.     See  p.  1S2,  Ex.  201. 


N 


W- 


87" 

so" 


71° 


44' 


SO" 

so 


16      41 
Ans.  1  hr.  6  min.  44  sec. 
2.  At  a  place  whose  longitude  is  35°  E.,  the  time  is  7  a.m. 
Monday.     What  is  the  longitude  of  a  place  where  it  is  6  p.m. 

Sunday  ? 

We  locate  the  place  36°  E. 
Since  6  p.m.  Sunday  is  earlier,  the 
_  other  place  must  be  farther  west. 

^ i \^ 1 E  We  locate  it. 

From  6  p.m.   Sunday  to   7   a.m. 
Monday,  is  13  hours  ;  13  hr.  =  195°. 
^'  See  p.  132,  Ex.  202. 

The  other  place  is  195°  west  of  the  given  place,  or  160°  west  of  the  prime 
meridian,  or  160°  west  longitude. 

292 


P.M. 

Sxm 


S5' 


7 
A.M. 
.If*"" . 


MISCKLLANKOUS  293 

Drato  a  diagram  for  each  problem. 

3.  The  longitude  of  St.  Petersburg  is  30°  19'  east.  What  is 
the  difference  in  longitude  between  St.  Petersburg  and  Boston  ? 

4.  What  is  the  true  difference  in  time  between  Boston  and  St. 
Petersburg  ? 

5.  At  a  place  whose  longitude  is  35°  17' 25"  E.,  the  time  is 
7.30  A.M.     What  is  the  longitude  where  it  is  11.55  p.m.  ? 

6.  At  a  place  37°  25' 37"  east  longitude,  the  time  is  7.30  a.m. 
What  is  the  time  at  a  place  whose  longitude  is  37°  25' 37"  W.  ? 

Standard  Time.  —  In  1883,  the  standard  time  system  was 
introduced  by  the  railroads  and  adopted  by  the  people  of  the 
United  States. 

By  this  system,  time  meridians  15°  apart  have  been  established  ;  viz.,  the 
meridians  60°,  76°,  90°,  105°,  120°  west  from  Greenwich.  Places  within  7^** 
east  or  west  of  a  time  meridian  have  the  time  of  the  time  meridian.  Thus, 
the  time  between  two  places  differs  by  tchole  hours,  or  not  at  all. 

7.  What  is  the  difference  in  standard  time  between  Boston  and 

Chicago  ? 

Note.  —  Boston  has  the  time  of  the  meridian  of  75° ;  Chicago,  of  the  meridian 
ot90P. 

International  Date  Line.  —  In  traveling  completely  around 
the  earth  from  west  to  east,  one  appears  to  gain  a  day ;  in  travel- 
ing from  east  to  west,  to  lose  a  day. 

Hence,  vessels  in  sailing  around  the  earth  must  correct  their  apparent 
days.  It  is  customary  to  make  the  change  at  the  meridian  of  180°  longitude. 
Vessels  going  east,  count  the  same  day  twice ;  vessels  going  west,  skip  a  day. 
Thus,  if  it  is  Sunday  when  an  east-bound  vessel  reaches  the  meridian  of 
180°,  it  counts  the  next  day  Sunday  also  ;  if  it  is  Sunday  when  a  west-bound 
vessel  reaches  this  meridian,  it  counts  the  next  day  Tuesday.  For  this 
reason,  the  meridian  of  180"^  is  called  the  International  Date  Line. 

a  A  traveler  found  that  his  watch  was  gaining  time.  If  his 
watch  was  accurate,  was  he  traveling  east  or  west  ? 

9.  If  a  man  were  to  travel  around  the  earth  from  west  to  east 
in  101  days,  reckoning  by  local  time  at  the  various  places,  in 
how  many  days  would  he  actually  make  the  trip  ? 


294 


PROBLEMS  IN   PHYSICS 


PROBLEMS   IN   PHYSICS 

The  specific  gravity  of  a  substance  is  its  weight  divided  by  the 

weight  of  an  equal  volume  of  water. 

IlluBtration :  A  certain  volume  of  lead  weighs  34  lb. ;  the  same  volume 
of  water,  3  lb. ;  the  specific  gravity  of  lead  is  34  -h  8,  or  U.S.  That  is,  lead 
Is  11.3  times  as  heavy  as  water. 

la  A  bar  of  iron  weighs  38.5  lb. ;  an  equal  volume  of  water 
weighs  5  lb.     What  is  the  specific  gravity  of  the  iron  ? 

11.  What  is  the  weight  of  1  cubic  foot  of  the  iron  ? 

12.  A  block  of  wood  weighs  15  lb. ;  the  amount  of  water  that 
it  displaces  weighs  25  lb.     Find  its  specific  gravity. 

13.  What  is  the  weight  of  1  cubic  foot  of  the  wood  ? 

14.  A  rock  on  the  bank  of  a  river  weighs  2  T. ;  but  in  the 
water  it  weighs  only  1  T.  4  cwt.     What  is  its  specific  gi-avity  ? 

In  the  Fahrenheit  thermometer,  the  freezing  and  boilincc  points 
are  32°  and  212°,  respectively;  in  the  Centigrade,  0°  aud  100° j  in 
the  Reaumur,  0°  and  80°. 

Reduce  and  explain  by  diagram,  14**  F  to  C;  to  R. 


Boiling        .  218° 


Freezing 


lOQO 


W> 


1800F  =  tnooc=:800R 

10F=     |0C=  |OR 

18°  F=   1U0C=  80E 

When  the  thermometer  reads  14°  F, 
the  temperature  below  fr«H»zlnff  js  1S°  F, 
10°  C,  or  S°  R,  i.e.,  14°  F  =  -  U>°  C,  or 
-80R. 


14° 

15.  The  temperature  of  a  room  is  68°  F.     Find  the  temperature 
in  C ;  in  R. 

16.  Lead  melts  at  335°  C.     Find  its  melting  .point  in  F. 

17.  Silver  melts  at  800°  R.     Find  its  melting  point  in  C. 

la   Alcohol  boils  at  78°  C.    Find  its  boiling  point  in  F  and  in  R. 
19.   On  a  certain  day  the  temperature  fell  from  86°  F  to  5°  C. 
How  many  degrees  R  did  it  fall  ? 


MISCELLANEOUS  295 


ACCOUNTS 


Bookkeeping  is  a  systematic  method  of  recording  business  trans- 
actions. Many  or  few  books  may  be  used.  Accounts  may  be 
kept  with  many  or  with  few  things.  The  underlying  principle 
is  simple:  Everything  tohich  owes  the  proprietor  is  a  debtor  (Dr.); 
everything  the  proprietor  owes  is  a  creditor  (Cr.). 

From  this  general  rule,  special  rules  may  be  derived : 

Persons.  Dr.  when  he  owes  me,  or  I  get  out  of  his  debt;  Cr. 
when  I  owe  him,  or  he  gets  out  of  my  debt. 

Cash.     Dr.  all  cash  receipts;  Cr.  all  cash  disbursements. 

Expense.    Dr.  all  outlays  from  ichich  no  direct  return  is  expected. 

Proprietor.  Dr.  everything  withdrawn;  Cr.  everything  put  into 
the  business. 

In  the  following,  only  two  books  are  used,  —  the  daybook  and 
the  ledger;  accounts  are  kept  with  only  four  things,  —  Cash, 
Expense,  the  Proprietor,  and  Persons.  At  the  time  of  each  trans- 
action, William  Harris  or  his  clerk  entered  it  on  the  daybook,  as 
on  p.  29G;  each  day  the  transactions  were  posted  from  the  day- 
book to  the  ledger,  as  on  p.  297. 

20.   Enter  the  following  in  daybook  and  ledger. 

Jan.  1,  1897.  William  Harris  began  a  grocery  business  with  $2000  cash 
and  9  2000  uidse.  I'aid  rent  for  January  by  bank  draft,  No.  500,  9  60.  Books 
and  stationery,  9  lo.     Cash  sales,  9&. 

Jan.  2.  Sold  A.  I).  Ku.s.sel,  on  account :  300  lb.  sugar  @  O-J ;  25  lb. 
coffee  @  30#  ;  18p  cwt.  flour  @  $2.50.     Cash  sales,  $07.60. 

Jan.  4.  Bought  of  A.  I).  Hussel,  on  %  :  horse  and  wagon  for  use  in  the 
business.  9  175.     Cash  sales,  $25. 

Jan.  5.  Rec'd  from  A.  D.  Kussel,  check  on  First  Nat'l  Bank,  for  $250, 
to  apply  on  his  %.     With<lrew  for  private  uae  $  100.     Cash  sales,  $  10.60. 

Jan.  0.  Sold  A.  1).  Kus.Hel  25  lb.  pork  @  11#;  5  gal.  molasses  @  25f ; 
5  cwt.  flour  (ft)  $2.80,  and  accepted  in  payment  an  order  on  Geo.  Adair,  for 
$  18,  which  Adair  paid  in  cash.     Vwh  sales,  $22.50. 

Jan.  7.    Paid  cash  for  I  T.  coal  for  store,  $  5.    Cash  sales,  $  80. 


296 

Datbook 


ACCOUNTS 
Jam.  1,  18S7 


Wm.  Harris  htgan  a  ffroeerf  bmsiHeas  with 

\ 

— 

i 

Cask, 

2000 

00 

V 

Mdse., 

2000 

00 

4O00 

00 

iV 

Paid  Jam.  remt  6y  bank  drqft  No.  S60, 

50 

00 

il 

B«»ka  amd  StaSioturf, 

15 

00 

65 

00 

i 

CatkSmUM, 

s. 

5 

00 

i 

Sold  A.  D.  Bnssel,  on  %  : 
300  lb.  sugar,                                    §  .06% 
S5  lb.  cogtt,                                           .SO 

20 

7 

00 

so 

180  ewL  /lour,                                     f.50 

450 

00 

477 

50 

i 

(UukSaUs, 

4- 

€7 

SO 

a 

BoL  of  A.  D.  Mussel,  on  %  : 
Horse  amd  Wagon  for  use  im  the  busimess. 

175 

00 

i 

Cask  Sales, 

5. 

25 

00 

a 

Ree'd  check  finym  A.  D.  Bussel,  am  %, 

250 

00 

n 

Withdrew  for  priwaU  use. 

100 

00 

V 

Cask  Sales, 

16 

50 

i 

6, 
Sold  A.  D.  Bussel,  for  order  <m  Geo.  Adair, 
whick  Adair  kas  tasked, 
tS  lb.  pork,                                       f  .11 
5gaL  wtotasses,                                      .25 

2 

1 

75 
25 

5  aet.  Jlour,                                           2.80 

14 

00 

IB 

00 

i 

Cask  Sales, 

22 

60 

W 

Bot.  for  caAj  to  use  im  store,  1  T.  coal. 

5 

00 

V 

Cask  Sales, 

7. 

SO 

00 

V 

Found  »H  gaim  to  daU  to  be 
Carrg  to  Cr.  side  j/roprietor's  %. 

€7 

00 

MISCELLANEOUS 


297 


Dr. 


Wm.  Harris 


Or, 


Jam. 


Pr9$ent  Worth 


296 


r   I 
ioo\oo 

396700 


4067  00 


Jan. 


Jan. 


Net  Gal* 
Preteut  Worth 


g$6 
296 


4O00  00 
67  00 


4067  00 


6967 


00 


COMk 


.lam 

M 

M 

U 

M 

m 

m 

m 

m  ■ 

— 

_ 

Jam. 

7 

Baiamee 


296 
296 
296 
296 
296 
296 
296 
296 


Baiaaea 


296 
296 
296 


65\00 

ioo\oo 

5  00 
IT»  #• 

2264  50 


14t4   *• 


2464  SO 


Expense 


Jam. 

1 

296 

6S0O, 

1 

« 

4 

296 

17Sm 

1 

M 

7 

296 

SOO 

t4*  ••! 

1 

A.  D.  Bmssel 


Jan:  S 


296 


477  6(A  Jam. 


296 
296 


nsoo 


250 

4tS 


00 


NoTS.  — To  find  the  proprietor's  worth,  find  the  total  resovrcee  and  UabiBtics, 
and  take  the  differeoee.  At  this  tkmt  the  ledger  ihovs  rcaonscce:  GMb,  %7»%J^ 
and  A.  D.  Bmeel.  f  52JS0.  The  isTeMory  of  Mdae.  fovad  hjr  cethMrtiaK  the  rahw 
of  goods  OB  hand  is  flTSO.  Theae  reaooreee  aggregate  $4067.  As  there  are  do 
HahiUtiea,  this  voold  be  the  proprietor's  preeent  worth  had  he  sot  withdrawn 
•100.  The  net  gaia  of  $67  to  carried  to  the  credit  aide  of  proprtolor'saeet..  and 
tweea  the  sides  shows  hto  prenot  worth  to  ha  $3987. 
\j  he  balaaeed,  as  siMNm  In  Gash  sect.,  wh«n  tlker  htemmb  rtrj 
large.    ThesBiall  type  footings  are  written  with  a  peacfl- 


298  ACCOUNTS 

21.  Turn  to  p.  297.  How  much  did  Wm  Harns  invest  ?  How 
much  did  he  withdraw  up  to  Jan.  7  ? 

22.  What  was  the  amount  of  cash  on  hand  Jan.  7  ?  Can  the 
Cr.  side  of  "  Cash  "  exceed  the  Dr.  ? 

23.  What  were  the  expenses  to  Jan.  8  ?  How  much  did  A.  D. 
Russel  owe  Jan.  7  ? 

24.  How  is  the  proprietor's  worth  found  ?  How  is  the  net  gain 
or  net  loss  found  ? 

25.  Continue  the  daybook  of  p.  296,  entering  the  following 
transactions. 

26.  Continue  the  ledger  of  p.  297,  posting  from  daybook  just 
completed ;  balance  the  accounts  as  before. 

27.  Make  out  and  receipt  Henry  Cook's  bill. 
2a   Make  out  A.  D.  Russel's  bank  check. 

Jan.  8.  Sold  Henry  Cook,  on  %  :  25  lb.  sugar  @  6^ ;  6  cans  peaches  @ 
121  ^  ;  2  bu.  potatoes  @  80^.     Cash  sales,  $27.30. 

Jan.  9.  Bot.  for  cash  :  3  bbl.  sugar  @  $25  ;  2  bags  coffee,  200  lb.  @  20f  ; 
3  cases  canned  goods,  6  doz.  at  $  1.20. 

Jan.  11.  Sold  A.  D.  Russel,  on  %  :  7  jars  pickles  @  25^  ;  14  doz.  clothes- 
pins @  5  <^ ;  2  lb.  candy  @  10  f.    Cash  sales,  $  34. 

Jan.  12.  Sold  A.J.  Davis,  on  %  :  20  lb.  sugar  @  6i  f  ;  17  lb.  beans  @  5f ; 
3  lb.  cheese  @\h9\  15  lb.  dried  beef  @  18^.     Cash  sales,  $23.25. 

,Ian.  13.   Bot.  oflBce  desk  and  fixtures  for  $  30. 

Jan.  15.  Sold  A.  D.  Ru.ssel,  on  %  :  20  lb.  beans  @  bf  \  15  lb.  coffee  @ 
30^  ;  14  cans  tomatoes  @  10^.    Cash  sales,  $  19. 

Jan.  16.  Bot.  for  cash :  4  kegs  fish  @  $3 ;  1  keg  vinegar,  20  gal.  @  15^ 
Cash  sales,  $  18. 

Jan.  18.  HecM  from  A.  D.  Russel,  on  %,  check  for  $100.  Sold  Henry 
Cook,  on  %  :  3  gal.  vinegar  @  25^  ;  20  lb.  sugar  @  8»  }> ;  2  lb.  tea  @  Qbf. 
Cash  sales,  $21.15. 

Jan.  19.  Withdrew  for  private  use,  $25.     Cash  sales,  $.34.50. 

Jan.  20.  Rec'd  from  Henry  Cook,  cash  in  full  of  his  %.  Sold  Jas.  Otis, 
on  %  :  50  cwt.  flour  @  $  3  ;  5  gal.  kerosene  @\bf\  1  doz.  cans  apples  @  15  f. 
Cash  sales,  $30.     Found  net  gain  from  Jan.  7  to  Jan.  21,  $100. 


MISCELLANEOUS 


299 


FORM   OF   BILL 


Emporia,  Kans.,  Jan.  5,  l<s97. 

^oU  to  a.  Jb.  Rua^C, 
Terms :  Cash.  1018  Market  Sti'fH. 


1897 


i5  Uy.  JK>vh  @  ffi 

5  e^.  fUwv  ®  f  2.80 
R&t/ci  ^a^yK&nt, 


i 

75 

/ 

25 

/^ 

00 

/8 

00 


Zihn,  /'fawtQ,. 


FORM  OF   BANK   DRAFT 


jYo.  ^ 


60. 


2ri)t  iFirgt  National  13ank  of  Emporia. 

Emporia,  Kans.,  fa/rv.  f ,   189  7. 

Pay  to  the  order  of lAh>v  /ifaviui-. 

J50.00.^^.^^.^.^.^.^<^t^.. 


To 


FirH  National  Bank, 
Kansas  City,  Mo. 


Dollars. 

MN) 

Cashier. 


FORM   OF   BANK  CHECK 


'..y\         '  Emporia,  Kans.,  fam,.  5,  189  7 

y  STfjt  iFirgt  National  93ank. 

'^^  Pay  to  the  order  of KMtv  /ifawU^^.^^^.^ 

s^.^^^'^.^^.^.^^otu^  humdv&cC  fvfty ^Dollars. 

a.  Jb.  Ruf,^. 
f  250.00. 


300 


GOVERXMEXT  DIVISIONS  OF  LAND 


GOVERNMENT   DIVISIONS  OF  LAND 

In  the  western  states,  the  laud  has  been  divided  by  the  United 
States  government  into  squares  6  mi.  each  way,  called  townships; 
each  township  has  been  divided  into  squares  1  mi.  each  way, 
called  sections;  many  sections  have  been  divided  into  4  equal 
squares,  called  quarter  sections;  and  some  quarter  sections  have 
been  divided  into  4  equal  squares,  called  quarter  quarter  sections. 


ATOWN8HIP 


A  SECTION 


6 

5 

4 

3 

? 

1 

7 

8 

9 

10 

11 

12 

IS 

17 

1« 

15 

14 

13 

19 

W 

n 

at 

S3 

M 

SO 

» 

» 

V 

» 

e 

31 

s 

S3 

3i 

» 

31 

K.iSecUoa 
(SMA.) 


S.W.J 
(160  A.) 


of   B-i-i 
.K-jsTT 


The  sections  forming  a  township  are  numbered  from  the  N.  E.  comer  as 
shown  in  the  left-hand  diagram. 

The  subdivisions  of  a  section  are  named  as  shown  in  the  right-hand  dia- 
gram.    A  division  smaller  than  a  quarter  quarter  section  is  called  a  lot. 

29.  Draw  a  north  and  south  line  to  represent  1  mi.  Upon  this 
line  construct  a  square,  and  divide  it  into  2  equal  parts  by  an 
east  and  west  line.  Upon  the  south  part,  write  its  name.  How 
many  rods  long  is  a  half  section  ?  How  many  rods  wide  ?  How 
many  acres  does  it  contain  ? 

30.  Divide  the  N.  \  section  into  2  equal  squares  by  a  north  and 
south  line.  Upon  the  N.  E.  square,  write  its  name.  ^  What  are 
the  dimensions  in  rods  of  a  quarter  section  ?  How  many  acres 
does  a  quarter  section  contain  ? 

31.  Divide  the  N.  W.  \  section  into  2  equal  parts  by  a  north  and 
south  line.  Upon  the  east  part,  write  its  name.  What  are  the 
dimensions  and  the  area  of  a  half  quarter  section  ? 

32.  Divide  the  west  half  of  the  N.  W.  quarter  into  2  equal 
squares  by  an  east  and  west  line.  Upon  each  part,  write  its 
name. 


MISCELLANEOUS 


301 


DEFINITIONS   AND   INDEX 
NoTB.    Z,  angle;  ±,  perpendicular;  A,  triangle;  O,  circle;  ||,  parallel. 


Acute  angle 

Acute-angled  triangle 

Addends  

Addition 

Agent 

Aliquot  part 

Altitude  I 

Amount ,  -I 

Analysis 

Angle 

Annual  interest 

Antecedent 

Apothem 

Arabic  notation 

Arc 

Are 

Area 

Arithmetic 

Assessment 

Average 

Bank  discount 

Base J 

Bonds  

Bookkeeping 

Brokerage . .  

Cancellation  — 

Chord 


250 
251 
16 
16 
198 
116 

253 
261 
16 
190 
216 
180 
249 
216 


170 
253 

10 

254 
138 
121 
7 
213 

58 
228 
145 
190 
253 
200 
212 

296 

207 

72 

254 


DBFINITION 


An  Z  less  than  90°. 

A  A  having  3  acute  A. 

Terms  in  Eiddition. 

I*roce8s  of  uniting  numbers  into  one. 

One  who  transacts  business  for  another. 

A  number  contained  an  integral  number 

of  times  in  a  given  number. 
A  ±  from  the  vertex  opposite  the  base,  to 


principal  plus 


Sum  of  several  numbers 
interest. 

A  process  of  solving  problems. 

The  opening  between  two  lines  which  meet. 

A  conception  of  interest  by  which  prin. 
and  int.  on  prin.  at  end  of  each  year, 
bear  interest. 

First  terra  of  a  ratio. 

Perpendicular  from  the  center  to  one  side 
of  a  regular  polygon. 

Method  of  expressing  numbers  by  the  char- 
acters, 0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 

Part  of  the  circumference  of  a  O. 

Unit  of  metric  land  measure. 

Number  of  square  units  in  a  surface. 

Science  of  numbers. 

A  tax  on  stockholders  to  meet  losses  or 
expenses. 

The  mean  of  several  unequal  quantities. 

Int.  on  amount  due  on  a  note  at  maturity. 

A  quantity  used  several  times  as  an  addend 
or  as  a  factor ;  that  on  which  %  is  com- 
putet! ;   the  part  on  which  a  figure  rests. 

Written  contracts  under  seal  to  pay  cer- 
tain sums  at  specified  times. 

Art  of  recording  business  tran-sactions. 

Commission  charged  by  a  broker. 

Dividing  both  dividend  and  divisor  by  the 
same  number. 

A  straight  line  connecting  two  points  in  a 
circtimference. 


302 


MISCELLANEOUS 


TKRM 


Cipher 

Circle 

Circulating  decimal 

Circumference 

Coefficient 

Commission 

Common  denominator 

Common  factor 

Common  fraction 

Common  multiple . 

Complex  decimal  

Complex  fraction  

Composite  number 

Compound  fraction 

Compound  interest 


Compound  proportion  — 

Compound  ratio 

Cone 

Consequent 

Convex  surface  of  a  solid . . 
Creditor 

Cube I 

Cube  root 

Curved  line 

Curved  surface 

Cylinder 

Debtor 

Decagon 

Decimal 


DBFINrriON 


40 
252 


110 
263 
145 

108 
88 

72 


76 

105  I 

83  ! 
i 
68 

83 
216 

174 

174 
260 
170 
261 
295 
238 
260 
244 
249 

251 

260 
295 
252 
103 


Naught;  zero. 

A  plane  fi^re  bounded  by  a  curved  line 

every  point  of  which  is  equally  distant 

from  a  [>oint  within  called  the  center. 
Decimal  containing  a  repeteiid. 
The  boundary  line  of  a  O. 
A  number  8howin<2:  how  many  times  a  base 

is  used  as  an  addend. 
An  agenfs  pay  for  buying  or  selling. 
A  denominator  common  to  two  or  more 

fractions. 
A  factor  common  to  two  or  more  numbers. 
A  fraction  whose  denominator  is  wriiteu 

below  a  horizontal  line. 
A  number  which  will  exactly  contain  eacli 

of  several  numbers. 
A  decimal  ending  in  a  common  fraction,  a^ 

A  fraction  having  fractions  in  one  or  bot'.i 
of  its  terms. 

A  number  having  another  set  of  factors 
besides  itself  and  1. 

A  fraction  of  a  fraction. 

A  conception  of  interest  by  which  prin.. 
int.  on  prin.  at  end  of  each  jr.,  and  all 
other  int.,  bear  interest. 

A  proportion  having  one  or  both  of  its 
ratios  compound. 

A  ratio  of  a  ratio. 

A  pyramid  whose  base  is  a  O. 

Second  term  of  a  ratio. 

All  its  surface  except  the  bases. 

One  to  whom  something  is  owing. 

Product  of  three  equal  factors  ;  a  solid 
having  its  three  dimensions  equal. 

One  of  the  three  equal  factors  of  a  number. 

A  line  which  constantly  changes  its  direc- 
tion. 

A  surface  such  that  no  two  points  lie  in 
same  plane. 

A  prism  whose  base  is  a  O. 

One  who  owes  a  debt. 

A  polygon  of  ten  sides. 

A  fraction  written  with  the  aid  of  a  deci- 
mal point. 


DEFINITIONS 


303 


TKRM 

p. 

103 
108 

DRFINITION 

Decimal  fraction 

A  fraction  whose  denominator  is  10,  100, 

Decimal  point       

1000 

A  period  written  before  tenths  in  a  deci- 

Denominate number 

Denominator 

Diagonal 

119 

83 

253 

254 

28 
1!)0 
21)1' 
207 

44 

207 

44 

44 

200 
299 

223 

223 
2G0 

159 
231 

261 

87 

239 

146 

170 

260 

68 

68 

mal. 
A   number  which  answers  the   question 

'  how  much  ? ' 
The  part  of  a  fraction  which  is  a  divisor ; 

the  number  showing  into  how  many  parts 

a  unit  is  divided. 
A  line  in  a  polygon  joining  two  vertices 

not  adjacent. 
A  line  through  the  centor  of  a  O  joining 

two  points  of  the  circumference. 
Result  obtained  in  subtraction ;  base  minus 

percentage. 
A  deduction  from  the  face  value  of  a  com- 
mercial paper ;  the  amount  the  market 

value  of  a  stock  is  below  the  par  value. 
A   number    to   be  divided ;    earnings  of 

stock. 
Process  of  finding  the  other  of  two  num- 

Diameter  

Difference 

Discount I 

Dividend | 

Division 

Divisor 

bers  when  one  of  them  and  their  product 
are  given. 
A  number  by  which  to  divide. 

Dodecahedron 

A  solid  bounded  by  twelve  faces. 

Draft 

A  written  order  directing  one  person  to 
pay  a  sum  of  money  to  another. 

The  person  to  whose  order  a  draft  or  note 
is  to  be  paid. 

The  person  who  signs  a  draft  or  note. 

The  intersection  of  two  faces  of  a  poly- 
gon. 

An  expression  of  equality. 

Process  of  finding  an  average  time  at  whit  1: 
several  payments  may  be  justly  made. 

A  A  whose  sides  are  equal. 

Fractions  equal  in  value. 

Process  of  finding  a  root  of  a  number. 

A  number  denoting  how  many  times  the 
base  is  used  as  a  factor. 

First  and  last  terms  of  a  proportion. 

Its  bounding  planes. 

An  exact  divisor  of  a  number. 

Drawee 

Drawer 

Edee 

Equation 

Equation  of  payments 

Equilateral  triangle 

Equivalent  fractions 

Evolution 

Exponent 

Extremes 

Faces  of  a  solid 

Factor 

Factoring 

Process  of  finding  numbers  whose  product 
is  given. 

304 


MISCELLANEOUS 


F. 

DEFINITION 

7 

Symbols  representing  numbers ;  diagrams 
representing  forms. 

Fraction 

83 

An  expression  of  division   in  which  the 

dividend  is  written  above  and  the  divisor 

below  a  horizontal  line  ;  one  or  more  of 

the  equal  parts  of  a  unit. 

Pmstuni           

200 

The  i>ortion  of  a  cone  or  a  pyramid  in- 
cluded between  its  base  -and  a  plane  ||  to 

the  base. 

Grace,  days  of 

228 

An  allowance  of  three  days  after  a  note  is 
due  for  its  payment. 

Gram 

141 

Metric  unit  of  weight. 

Greatest  common  divisor. 

U 

The    greatest  number   tlmt    will   exactly 
divide  each  of  several  numbers. 

Heptagon 

252 

A  polygon  of  seven  sides. 

Hexagon  

252 

A  polygon  of  six  sides. 

250 

A  straight  line  II  to  the  horizon. 
The  side  of  a  right-angled  A  opposite  the 
right  angle. 

253 

Icosahedron  — 

260 

A  solid  bounded  by  twenty  faces. 

Improper  fraction 

84 

A  fraction  whose  value  is  1,  or  more  than  1. 

Indorsement 

223 

The  signature  on  the  back  of  a  note. 
Agreement  to  pay  for  loss  or  damage. 
A  whole  number. 

Insurance  

20i 

Integer  

7 

Interest  

21« 

Money  paid  for  the  ilsc  of  money. 
Boundary  line  between  regions  where  the 

International  date  line 

293 

calendar  day  is  different,  coinciding  ap- 

proximately with  meridian  of  180°. 

Involution  

2:w 

Process  of  finding  a  power  of  a  number. 
A  polygon  whose  angles  are  unequal. 

Irregular  polygon 

252 

Isosceles  triangle 

251 

A  A  having  two  of  its  sides  equal. 

Least  common  multiple  . . 

77 

The  least  number  that  will  exactly  contain 
each  of  several  numbers. 

Least  common  denomina- 

76 

The  least  common  multiple  of  several  de- 

tor 

nominators. 

Liabilities 

297 
249 
140 

Debts  to  be  paid. 

That  which  has  one  dimension. 

Line 

Liter 

Metric  unit  of  capacity. 
A  quantity  represented  by  letters. 
Division  in  which  processes  are  written  out. 
Distance  E.  or  W.  of  the  prime  meridian, 
as  measured  on  the  equator. 

Literal  quantity 

145 

Long  division  

58 

Longitude 

292 

Market  value 

207 
22.] 

Price  of  stock  in  the  market. 

Maturity  of  a  note 

Date    at   which  a  note   legally  becomes 

due. 

DEFINITIONS 


305 


TKBM 

F. 

DEFINITION 

Mean  proportional 

Mffin^f                    

170 

170 
68 
60 
249 
186 
136 

28 
145 

83 
68 
76 
35 
36 

35 
145 
198 
262 
7 
223 

7 
84 

7 
68 

68 

250 
250 

260 
261 
262 
260 
8 
249 

One  of  the  means  of  a  proportion  in  which 

the  means  are  equal. 
Second  and  third  terms  of  a  proportion. 
Factors. 

Parts  of  equations  connected  by  the  sign  =. 
Science  of  measurement. 
Metric  unit  of  linear  measure. 

Measures  of  numbers 

Members 

Mensuration  

Meter   

Metric  system 

Minuend  . .    

Minus  sign 

Mixed  number 

A  decimal  system  of  measurement,  whose 
principal  unit  is  the  meter. 

Number  in  subtraction  to  be  diminished. 

Sign  which  denotes  subtraction,  or  the  op- 
posite of  '  +.' 

An  integer  plus  a  fraction. 

Number  wh  ch  contains  another  an  integral 
number  of  times. 

Number  to  be  multiplied.  . 

Process  of  finding  the  sum  when  the  same 
number  is  used  several  times  as  an 
addend. 

Number  by  which  to  multiply. 

A  quantity  preceded  by  the  minus  sign. 

Sum  left  after  every  charge  is  paid. 

A  polygon  of  nine  sides. 

Art  of  writing  numbers  by  symbols. 

Written  promise  to  pay  a  sum  of  money 
at  a  stated  time. 

Unit  or  collection  of  units. 

Multiple 1 

Multiplicand 

Multiplication 

Multiplier 

Negative  quantity 

Net  proceeds 

Notation    

Note  ... 

Number   

Numerator 

The  part  of  a  fraction  which  is  the  divi- 
dend ;  the  number  showing  how  many 
parts  are  taken. 

Process  of  naming  or  reading  numbers. 

Numbers  having  no  common  factor  greater 
than  1. 

Numbers  prime,  each  to  each  of  the 
others. 

Line  neither  ±  nor  ||  to  the  horizon. 

An  Z  formed  by  two  lines  which  meet  so 
as  to  make  the  adjacent  A  unequal. 

An  Z  greater  than  a  rt.  Z. 

A  A  which  has  one  obtuse  Z. 

A  polygon  of  eight  sides. 

A  solid  bounded  by  eight  faces. 

One  of  the  three  places  in  a  period. 

Lines  in  the  same  plane  which  do  not  meet 
however  far  they  may  be  produced. 

Numeration 

Numbers  prime    to    each 

other 
Numbers  severally  prime. 

Oblique  line 

Oblique  angle 

Obtuse  angle 

Obtuse-angled  triangle. . . 
Octagon 

Octahedron 

Order    

Parallel  lines 

AMBR.  ARITH. — 20 


306 


MISCKLLANEOUS 


TKRM 

p. 

DRFIMITIOM 

Parallelogram 

251 

60 

206 
223 
223 
262 
100 
190 
253 
7 

250 
251 

145 

249 
202 

251 

200 
145 
238 

202 

227 

292 

68 

198 
216 

77 
260 

26 

A  quadrilateral  having  both  pairs  of  ius 
opposite  sides  parallel. 

Curved  marks  inclosing  one  or  more  quan- 
tities considered  as  a  whole. 

Face  or  nominal  value. 

Parenthesis 

Par  value 

Partial  payments 

Payee 

Payments  in  installments  of  notes,  etc. 

The  person  to  whom  a  note  is  payable. 

A  polygon  of  five  sides. 

Hundredths. 

Computations  with  per  cent. 

Distance  around  a  polygon. 

In  numeration,  groups  of  3  figures  each; 

in  evolution,  groups  of  any  number  of 

figures. 
A  line  which  meets  another  so  as  to  make 

the  adjacent  A  equal. 
A  surface  such  that  a  straight  line  joining 

any  two  points  in  it,  lies  wholly  within 

that  surface. 
Sign  which  denotes  addition  or  a  positive 

quantity. 
That  which  has  no  dimension. 

Pentagon  

Per  cent  . .       

Percentage 

Perimeter 

Periods 

Perpendicular  line 

Plane   

Plus  sign 

Point    

Policy 

A  contract  between  an  insurance  company 

and  the  person  insured. 
A    plane    surface     inclosed    by   straight 

lines. 
A  tax  levied  at  so  much  per  head. 
A  quantity  preceded  by  the  plus  sign. 
A  product  obtained   by  using  the  same 

number  several  times  as  a  factor. 
Sum  paid  for  insurance  ;  excess  of  market 

value  above  par  value. 
Present  value  of  a  debt  due  at  a  future 

Polveon 

Poll  tax 

Positive  quantity 

Power 

Premium  

Present  worth 

Prime  meridian  . .    . . 

time. 
The  circle  passing  through  the  poles  of  the 

earth  and  through  Greenwich,  Eng. 
A  number  which   has  no  integral  factor 

besides  itself  and  1. 
One  who  employs  an  agent;    a  sum  on 

which  interest  is  reckoned. 
A  fundamental  truth. 

Prime  number 

Principal | 

Principle 

Prism 

A  solid  having  its  two  bases  equal  and  || 
polygons,  and  its  faces  rectangles. 

An  example  in  which  the  operations  are 
not  stated. 

Problem 

DEFINITIONS 


307 


TBKM 

p. 

DBFIMITION 

Proceeds 

Product 

Proof  

228 

36 

79 

170 
200 
254 
251 

44 

254 

170 

251 

85 

252 

28 
48 
110 

251 
251 

250 

251 
15 

239 
251 
254 

254 

264 

254 
254 
206 

Difference  between  face  value  of  a  note 
and  the  discount.  See  "  net  pro- 
ceeds." 

Result  in  multiplication. 

Operation  checking  accuracy  of  a  calcula- 
tion. 

An  equality  of  ratios. 

A  solid  having  one  base,  and  its  faces  A. 

Quadrant     

A  sector  equal  to  ^  of  a  O. 

Quadrilateral   

A  plane  figure  bounded  by  four  straight 

lines. 
Result  in  division. 

Rftdius  

A  straight  line  drawn  from  the  center  of 

a  O  to  the  circumference. 
An  expression  of  division   in  which  tlie 

dividend  is  written  before,  and  the  divisor 

after,  a  colon. 
A  parallelogram  whose  A  are  all  rt.  A. 
Process  of  changing  the  form  of  an   ex- 
pression without  chanj^ing  its  value. 
A  polygon  which  is  equiangular  and  equi- 
lateral. 
The  result  in  subtraction  ;  the  part  left 

undivided  in  division. 
A  figure  or  set  of  figures  in  a  circulating 

decimal  which  repeat. 
A  parallelogram  whose  A  are  not  rt.  A. 
A  rhomboid  whose  sides  are  equal. 
An  Z  formed  by  two  lines  meeting  so  as  to 

make  the  adjacent  angles  equal. 
A  A  containing  1  rt.  Z. 
Method  of  expressing  numbers  by  means 

of  the  characters.  I,  V,  X,  L,  C,  D,  M. 
One  of  the  equal  factors  of  a  number. 
A  A  which  has  no  two  of  its  sides  equal. 
A  straight  line  cutting  the  circumference 

of  a  O  in  two  points. 
A  portion  of  a  circle  between  two  radii  and 

their  included  arc. 
A  portion  of  a  O  between  a  chord  and  ite 

arc. 
A  segment  which  is  )  of  a  O. 
A  sector  which  la  J  of  a  O. 
One  of  the  ecjual  parla  into  which  a  capital 

stock  is  divided. 

Ratio 

Rectangle 

Reduction 

Regular  polygon 

Remainder | 

Repetend 

Rhomboid 

Rhombus     

Richt  ancrle 

Right-angled  triangle  — 
Roman  notation 

Root 

Scalene  triangle 

Secant 

Sector 

Segment 

Semicircle           

SexUnt 

Share 

308 


MISCELLANEOUS 


F. 

DSriKITIOK 

Short  division 

61 

268 

216 
170 
261 

260 

160 

294 

260 

238 
251 
260 
240 
293 
139 
206 

28 

28 

16 

251 

254 

200 

60 

260 
294 
201 
159 

Division  in  which  the  operations  are  per- 
formed mentally. 

Figures  whose  A  are  mutually  equal  and 
whose  homologous  sides  are  propor- 
tional. 

Interest  on  the  principal  alone. 

An  equality  of  two  simple  ratios. 

Distance  from  vertex  of  a  pyramid  to  base 
of  one  of  its  faces. 

That  which  has  length,  breadth,  and  thick- 
ness. 

Process  by  which  the  answer  to  a  problem 
is  obtained. 

The  ratio  of  the  weight  of  a  substance  to 
weight  of  same  volume  of  water. 

A  solid  bounded  by  a  curved  surface,  every 
point  of  which  is  equally  distant  from  a 
point  within,  called  the  center. 

The  product  of  two  equal  factors  ;  a  rec- 
tangle whose  sides  are  equal. 

A  prism  whose  base  is  a  square. 

One  of  the  two  equal  factors  of  a  number. 

Similar  figures 

Simple  interest 

Simple  proportion 

Slant  height 

Solid 

Solution 

Specific  gravity 

Sphere 

Square 

Square  prism 

Square  root 

Standard  time 

Railroad  time. 

Stere 

Metric  unit  of  wood  measure. 

Stock 

Property  owned  by  members  of  a  charter 
company. 

Process  of  finding  the  other  of  two  num- 
bers when  one  of  them  and  their  sum 
are  given. 

A  number  to  be  subtracted. 

Subtraction 

Subtrahend 

Sum 

The  result  in  addition 

Surface 

That  which  has  two  dimensions. 

Tangent  

A  straight  line  which  touches  the  circum- 
ference of  a  O  at  only  one  point. 

A  sum  assessed  on  a  person  or  on  property 
to  meet  public  expenses. 

Expressions  separated  by  the  signs  '-I-*  or 
*  — ' ;  the  numerator  and  denominator 
of  a  fraction  ;  nomenclature  in  the  vari- 
ous operations. 

Solid  bounded  by  four  equal  A. 

An  instrument  for  measuring  temperature. 

Discount  from  the  list  price  of  goods. 

Transferring  a  term  from  one  member  of  an 
equation  to  the  other. 

Tax 

Terms 

Tetrahedron 

Thermometer 

Trade  discount 

Transposition 

DEFINITIONS 


309 


p. 

251 

261 

251 

260 
227 

84 
167 
260 
260 

60 

261 

DK7INITIOM 

TraDezium    

A  quadrilateral  having  neither  pair  of  its 

opposite  sides  ||. 
A  quadrilateral  having  one  pair  only  of  its 

Trapezoid 

Triangle 

Triangular  prism 

True  discount     

opposite  sides  II . 
A  plane  figure  bounded  by  three  straight 

lines. 
A  prism  whose  bases  are  i^. 
Principal  minus  true  present  worth. 
One,  a  single  thing. 
Quantities  whose  values  are  sought. 
Union  of  three  or  more  edges  of  a  solid. 
A  line  ±  to  the  horizon. 

Unit 

Unknown  quantities 

Vertex 

Vertical  line 

Vinculum 

A  straight  line  placed  over  two  or  more 
quantities  to  show  they  are  to  be  re- 
garded as  a  single  term. 

The  number  of  cubic  units  in  its  contents. 

Volume  of  a  solid 

ANSWERS 


pp  25  to  41 


pp.  41  to  41 


Addition 
PP.2&-27 

140.  2,0»),810 

141.  1<5.2»R),«:J4 

142.  l«»,o<>:^,43«) 

143.  I\),<nsri0o 

144.  mjfu.m) 

148.  S'-.'02 

149.  79  miles 

150.  1H2  pounds 

151.  »UG2 

152.  I.'i6  strokes 

168.  9t)(32r> 
164.  S21.()*H'. 

155.  S87:n 

156.  ()2:i8  trees 

157.  1545  miles 

158.  7:) ,<)}«)  men 

169.  10,2.S7,5:« 
square  miles 

160.  2«),002feet 

161.  4915  miles 

162.  2843  bushels 


Subtraction 
pp.  33-34 

85.  12<X);  by  200 

86.  S1400 

87.  (i«8 

88.  201 

89.  8222 

90.  10  spaces 

91.  20<>  acres 

92.  8200 

98.  87,590  .square 

miles 
94.  1007  miles 
96.  85056 

96.  204  trees 

97.  87536 

98.  460  miles 


Multiplication 
p.  41 
88.  fi,4'J4,:<02 
84.  HH5,715 
86.  ii,7l9,:«.s 


Multiplication 
p.  41 

86.  1,992,^8 

87.  7,714,620 

88.  7,180,9a5 

89.  5,178,576 

90.  8,300,<r76 

91.  8,173,492 

92.  6,0:W,749 

93.  (■),♦» k'»,.S*M» 

94.  7.2«}7,842 

95.  4,793,633 

96.  7,279,272 

97.  3,289,75() 

98.  I,4(i5,614 

99.  1,8(59,240 

100.  1,.'>03,62.') 

101.  2,201,892 

102.  l,124,(Ui5 

103.  2,070,7:« 

104.  2<>,8.'i9,<X)0 

105.  57,827,000 

106.  21,7()5.100 

107.  2.^,575,000 

108.  2,86,'^000 

109.  28.519,200 

110.  19.8<t0.(XX) 

111.  .<7, 740,000 

112.  12.400,000 

113.  »J00,938,744 

114.  98,400,000 

115.  788,407,008 

116.  4«)1,.'W53,«>84 

117.  478,70<).43<> 

118.  742,9»«,9.')4 

119.  41fi,(r>9,.^4 

120.  «)82.880,000 

121.  431,257,7r»<> 

122.  212,536,«r)4 

123.  442,928,901 

124.  2.'^,.■»4,^>.')4 
126.  ri04,849,8(»8 

126.  620.919,a50 

127.  621. aw.' "OS 

128.  490,119.007 

129.  .'>,'>5,«»88,«W8 

130.  .•«5,580,:{94 
181.  3i;7.W>8,400 

132.  6V;.8.''.9.0»)8 

133.  l(M;,:vk').(v4s 

134  4;<,48(),'.t90 


pp.  41  to  54 

Multiplication 
pp.  41-43 
136.   48<J,978,700 

136.  223.0(;4,a'SO 

137.  21.'),747,3i>2 

138.  226,94<;,000 

139.  1(>1»,'»*.W»,126 

140.  8<),079,021 

141.  2;i2..')78,200 

142.  l();i,<)19,026 

147.  S:K)24 

148.  41,0<)2,000 
stitches 

149.  S586 

150.  226,176  letters 

151.  91,289,79() 
miles 

152.  70,215,270.400 
cubic  vards 

153.  S  16.978 
164.   S4«)81 

155.  146,289  miles 

156.  810,285 


Division 


pp.  63-64 

128. 

364;  6:189; 

21 

124. 

138:364;  18 

126. 

215;  5:476; 

12 

126. 

711;  9:2.')2 

127. 

5:i8;  2:408:3 

128. 

1875:367;  3 

129. 

5002;  4:3<;9; 

10 

130. 

4000; 4;  322 

181. 

2011 :  15: 

262;  8 

132. 

3179;  23: 

169;  12 

136. 

2 

137. 

4 

138. 

5 

139. 
140 

141 
142 
143. 


9 

19;  206 
26;  156 
!!•;  19f> 


pp.  64  to  66 

Division 
pp.  64-66 
144.    12;  49 
146.   18;  187 

146.  4 

147.  6 

148.  8 

149.  5 

150.  9 

166.  30mH 
156.  175tXSll 

167.  210il8» 

168.  90]i;:i 

169.  3089;  30 

160.  4362;  1 

161.  2931 

162.  .•082 

163.  4107 

164.  2573;  62 

165.  3871 

166.  38,413 

167.  29,615;  J 

168.  24.542 

169.  45,684 

170.  41,075 

171.  90,009 

172.  80,808 

173.  l.'iJW 

174.  2009 
176.  806 

176.  374 

177.  423 

178.  :«33 

179.  3949 

180.  31 

181.  200 

182.  220 
188.  \ri;  103 
184.   106 
186.   237 

186.  45 

187.  468 

188.  387 

189.  267;  2076 

190.  214 ;  .T254 

191.  25;  T.m 

192.  61;  176.3 

193.  1.^;  2849 
317 

125;  1000 
218:  2008 


797 
1«>36 


194 
196 
196 


811 


312 


AiNSWERS 


pp.  56  to  66 


pp.  66  to  71 


pp.  71  to  81 


pp.  82  to  90 


Division 
pp.  5(>-69 

197.  476;  14134 

198.  313;  12988 

199.  127;  15333 

200.  117;  899 

205.  81300 

206.  60  people 

207.  64  bushels 

208.  8136ilt 

209.  S26 

210.  85500;  83000 

211.  6  years 

212.  113  days 

213.  550  sacks 

214.  12  miles 

215.  5  farms,  100 
acres 

216.  82 

217.  168  pp. 

218.  3(>i  months 

219.  823 

220.  65  times;  830 
acres  remain- 
ing 

221.  226 

222.  102  acres 

223.  8152 

224.  160  tons 

225.  18  cents 

226.  (>4  hours 

227.  15  feet 

Operations  Com- 
bined 
pp.  61-66 

13.  11 

14.  35 

15.  11 

16.  3 

17.  11 

18.  4 

19.  36 

20.  8 

21.  30 
23.  58 

51.  39a5  votes 

52.  3(«9  votes 

53.  8  payments 

54.  836150 

55.  12  miles 


Operations  Com- 
bined 

pp.  66-67 

8  miles 
10  hours 

2  miles 

3  hours 
810 

8  weeks 
27  sheep,  8 13 
left 
82022 
81680 
8  308 
8403 

67.  85470 

68.  8  2072 
2262  rods 
400  feet 
3  days 
30  boys 
A.9f;  D.3# 


56. 
57. 
58. 
59. 
60. 
61. 
62. 

63. 
64. 
65. 
66. 


70. 
71. 
72. 
73. 


74.   81479 

Factoring 

pp.  70-71 

29.  2,2,2,5,3,3, 


30. 


31. 


48 


44 


45 


106 
107 
108 
109 
110 
111 
112 
113 
114 
115 
116 
117 
118 
119 
120 
160 
161 
162 
163 
164 
165 
169 
170, 


34. 


7,  11;  4,  6 
9,  10,  12,  14, 
15,  18,  20,  21, 
22,  24,  28,  30, 
33,  35,  36,  40, 

42,  44,  45,  55, 
56,  00,  63,  66, 
70,  72,  77,  84, 
88,  90,  S« 
2,3,3,11,89; 
6,33.9,22,66, 
99,18 
5,  13 ;  65 
1,2,3,5,7,11, 
13,  17,  19,  23, 
29,  31,  37,  41,! 

43,  47,  53,  59,   179 
61,  67,  71,  73,  i 

79,  83,  89,  97  ; 
26  numbers     '180 
Yes,  no,  yes,  j  181, 
yes,  yes,  no     1 182 


175 
176 


Factoring 
pp.  71-81 
409 

2.  2,  2,  3,  5 ; 
5,  11,  2,  2;  2. 
2,  2,  2,  3,  3,  3, 
13 

2,2,3,3,3;  2. 
2,2,2,3.3;  6, 
17,17 

2,2,  3,  13;  2, 
2,  2, 2, 2,6;  2, 
2.2,2,2,3,7, 
11 
,  3, 3, 3, 5;  3, 3, 
2, 2, 2, 2, 2 ;  5, 
5,  11,  11 
165 
95 
144 
106 
840 
626 
950 
17 
1 

13 
107 
13 
11 
9 
7 

14,400 
55,125 
217,728 
34,476 
43,890 
53,100 
1 

2,  3,  4,  5,  6, 7, 
or  8,  times  1 
2,  3,  4,  5,  6,  7, 
8,  or  9,  times  1 
2,  3,4,5,6,7, 
8,  or  9,  times  1 
12     feet     for 
ends,  16  feet 
for  sides 
60  yards 
25  feet 
4  feet 


Factoring 
p.  82 

184.  A  and  B,  24 
min.;  A  and 
C,  18  min.;  B 
and  C,  72  min 

185.  20  times 

186.  (iOcKfjs 

187.  63  eggs 

Common  Frac- 
tions 
pp.  85-90 

21.  iS;  i 

22.  i;  }i 

23.  f:  t 

24.  il;  }{ 

25.  ii;  f^ 

26.  !;  H! 

27.  15;  ii 

28.  ?lii;  H 

29.  A;  I 

39.  Ml 

40.  Itil 

41.  Mm 

42.  /AiV 

54.  ft'.,fA,m 

55.  m,i!l,  i» 

56.  t;:.  Ii},  lii 

57.  m\,^^,^^ 

58.  ilH,ifll,  mi 

66.  23/, 

67.  64il 

68.  55H 

69.  :mn 

70.  152  AV 

71.  80SI 

77.  W;  W 

78.  ^?i*;  *f|^ 

84.  12838 

85.  l«v)SS 

86.  53^ 

92.  II;  40}» 

93.  lii;  45/% 

94.  II;  mn 

107.  zh 

108.  2079 

109.  i 

110.  i 

111.  A 

112.  } 


ANSWERS 


818 


pp.  90  to  94 


pp.  96  to  100 


pp.  100  to  106         pp.  106  to  111 


Common  Frac- 
tions 


pp.  90-94 


lis.  6 

114.  5k 

115.  i 

116.  2 
165 


166.  1,^ 

167.  m 

168.  A 

169.  fr 

170.  8* 

171.  32 

172.  7J 

173.  22 

174.  31 

175.  lA 

176.  411 

177.  6i 

178.  6A 

179.  3i 

184.  62 A% 

185.  r>l»|J5 

186.  2()iii 

187.  52.Vo 

188.  81M 

189.  73IH 

190.  60!ii 

191.  68i( 
198.  23AV 

193.  21111 

194.  15<»;!1 

195.  2()2ai 

196.  15.V14 

197.  32(M5 

198.  178981 

199.  171003A 

200.  187<»/r 

201.  2H.'>12!| 

202.  ♦ViiM.'iJ 

203.  Av.mn 

204.  1801 
905.  31151 

206  umt 

207  WiUkt 

208.  2124 

209.  niH 


210 
211 


l.iUI 


Common 

Common 

Decimals 

Fractions 

Fractions 

pp.  lOG-lll 

pp.  1)6-100 

p.  100 

68. 

9    thousand. 

219.   1:  i 

270. 

A 

and    9   thou- 

220. U;  14 

271. 

108  sheep 

sand  372  ten- 

221.   i;  A 

272. 

12  apples 
i'*  of  whole 

millionths 

222.  1;  t. 

273. 

68. 

S  VXiH.\m 

223.   f 

ship 

69. 

S.374.(;!»3 

224.   1 

274. 

10;  6 

70. 

1?2H2:<.4016() 

225.   1 

275. 

120  apples 

71. 

68.2711 

226.  1 

276. 

«r,6 

72. 

13.1.351732 

227.  U 

277. 

:«5  cattle 

73. 

64.0255 

228.  5 

278. 

SI 

74. 

9821.631 

229.  9 

279. 

A,  30  years 

75. 

.'>4.864 

230.   10 

280. 

(K)  years 

76. 

.yas 

231.  19i 

281. 

5  days 

77. 

9i).13126 

232.  1 

282. 

r)0  yards 

78. 

768.04 

236.  i 

283. 

8  days 

79. 

1.478<)94 

287.  1 

80. 

408.5081 

238.  if 

81. 

.32.'«»775 

239    U 

242!  Sura,  39A; 

Decimals 

82. 
83. 

449.95854 
77477.4 

dif.,5H; 

p.  106 

84. 

.5021.75 

prod.,  3793; 

85. 

{»7.'-.3.76 

quo.,  IjVi; 

31. 

.000500 

86. 

248.^9.02736 

prod,  of  sum 

32. 

17.7 

87. 

.015 

and  dif., 

33 

.000006.38 

88. 

13.1489-1- 

233A«r 

34. 

.000000000023 

89. 

3.10(53  + 

243.   156584 

35. 

.000085002 

90. 

153.482 -h 

246.   A 

36. 

.7007 

91. 

.03()3-H 

247.  !t 

37. 

.oo;H)0 

92. 

.00633 -f 

251.  99^ 

38. 

.604 

93. 

6.7808-}- 

252.  54^ 

39. 

9431.0906532 

94. 

.0(K)0()01 

253.  24,25,81 

40. 

.0000! 

95. 

10,000,()00 

254.  $16 

41. 

2.000,000,000- 

96. 

.01599 -h 

266.  I 

.000,000,002 

97. 

1. 05997 -H 

256.  4 

42. 

8:wu).oa') 

9P. 

.01399  + 

257.  AV 

43. 

.OJ^^nX);) 

99. 

2,000,000 

258.  6  acres 

44. 

.()()100<'»<)04 

100. 

.000000176  + 

259.  3U  yards 

55. 

98  million  (VM 

108. 

tIi  :  tit 

260.   18  coats 

thou.sand   2«»S 

109. 

A:  •!< 

261.  41 

hundred -mil - 

110. 

A:  A 

868.  1  in  com 

lionths 

111. 

i4ii:  A 

863.  8  dozen 

66. 

8;J   million    8 

118. 

t4.;  411 

864.  24  years 

hundred -mil- 

lis. 

A:  A 

866.  $6 

lionths 

114. 

Ti.;4i 

866.  S 15000 

87. 

9  thousand  2 

116. 

4;  411 

267.  :i  apples 

hundred,  and 

116. 

sm 

268    <:ii 

929  ten-thou- 

117. 

441;  1 

269.  $-V, 

sandths 

118. 

AVb;  A 

314 


ANSWERS 


pp.  Ill  to  118         pp.  180  to  181         pp.  181  to  184         pp.  184  to  189 


Decimals 
pp.  111-118 

119.  /«:  xA^ 

120.  iV»:  hi 

121.  A:  ^Afc 

126.  .1H75;  .10 

127.  .117187:.;  .OUli 

128.  .7525 

129.  .87.i75 

130.  .(J00625 

131.  No;  13  is  a 
factor,  other 
than  2  or  5 

132.  .H44>A 

133.  .H4t>H- 

134.  .846163 

143.  nn 

144.  m 

145.  m 

146.  MAP. 

147.  7A 

148.  8ilf8 

149.  :K>iIi 

150.  ,V 

218.  S«4.71876 

219.  » 135.94 

220.  $(>.67 

221.  « 48.29 

222.  $54.69 
224.   S 64.19 

226.  $ti4,395.06; 
S64,31>5.0t)l 

227.  20  decimal 
places 

228.  234.63  X 
«(1.06)S,  or 
$  313.91) 

229.  S  64.68 

230.  420.16+times 

231.  2695.4928  feet 

232.  41.661  days 

233.  $.004  + 

234.  §  3(r).25  (5 
leap  years) 

235.  9.278 +bu. 
236    8000  pounds 

237.  $23.76 

238.  40(1.64  bu. 

239.  $218.24 

240.  $327.24 


Denominate 
Numbers 

pp.  130-131 

154.  225  pt. 

155.  7S46far. 

156.  1860  drops 

157.  10,079  gr. 

158.  12cwt.  561b. 

159.  110  rd.  2  yd. 
7iin.     ■ 

160.  88  .sq.  rd. 

161.  r.Oinin. 

162.  1  mo.  15  da. 
(30  da.  to  the 
mo.) 

163.  61  so.  rd.  18 
sq.  yd.  1  sq.  ft. 
501  sq.  in. 

164.  11  cu.  ft.  432 
cu.  in. 

165.  6  oz.  13  pwt. 
8gr. 

166  2  s.  9d.  3.936 
far. 

167.  lpk.2qt. 
1.024  pt. 

168.  '2M  rods  2 
yards  1  foot 
4  inches 

169.  7  cwt.  77  lb. 
12t  oz. 

170.  95  33  13 
3.904  gr. 

171.  lOP  42' 

172.  3qt.  1  pt. 

173.  30  .sq.  mi. 

177.  8  bbl.  6  gal.  1 
qt.  1  pt.  1  gi. 

178.  80Z.  6gr. 

179.  in>  5  5  73 

2  3  17  gr. 

180.  8eh.  2rd. 

181.  £1642  4s 

182.  234  rd.  1  yd. 
1  ft.  1  in.         ; 

183.  1  sq.  rod6sq. 
yards  2  sq. 
feet    83   sq. j 
inches 


Denominate 

Numbers 

pp.  131-134 


184. 

1  CO.  vd.  1  cu. 

ft.  10  cu.  in. 

185. 

iti  tti. 

186. 

iiioo  Tp. 

187. 

lU^K 

188. 

iV\AA>mi. 

189. 

2ii  ed. 

190. 

'211U  T. 

191. 

:u;2r.gal. 

192. 

r).H59:i75  bu. 

193. 

i:«;.425 

194. 

7.4<;01»526T. 

195. 

.112 +  C. 

196. 

5.41071  +  yr. 
by  1st  meth. ; 

6.378 +yr.  by 

2d  meth. 

203. 

7ty>i  lb. 

204 

5.5r.f  tb 

205. 

24  bu. 

206. 

21 J  hbl. 

207. 

2.9rK5-f-bbl. 

208. 

46igal. 

S09. 

•20  min.  20J 

sec.;  24  min. 

24  sec. 

210. 

76^  16'  15" 

214 

2.1  T.   8  cwt. 

M2  lb.  14  oz. 

215. 

27  rd.  2  yd.  1 

ft. 

216. 

27  lb.  3  OZ.  7 

pwt.  15  gr. 

217. 

51  yr.  10  mo. 

21  da.3hr. 

218. 

178  rd.   1  yd. 

2  ft.  6  in. 

219. 

178.3.31  rd. 

220. 

4r>2<).6  gr. 

221. 

Oct.  24;  Nov. 

13 

222. 

Oct.  26;  Oct.  6 

226. 

2  lb.  5  oz.  16 

pwt.  19  gr. 

227. 

10  rd.  1  yd.  2 

ft.  1  in.;  1  A. 

18  sq.  rd. 

240. 
241. 


Denominate  Num- 
bers 

pp.  134-139 

828.   mda.; 

.5291+ da.; 

12  hr.  41  min. 

Ml  sec. 
229.  213  da. 
280.  7  mo.  1  da. 
231.  2  da. 

235.  84^  64' 24" 

236.  25  bn.  3  pk. 
4  qt. 

237.  16  gal.  3  qt. 
1  pt. 

238.  41  rd.  1  yd. 

239.  30  da.  10  hr. 
59  min.  40  sec. 
17  8^.  yd.  to 
sq.  in. 

19  hr.  30  min. 
30  sec. 

242.  KiO  rd.  1  ft. 
1!  in. 

243.  1  T.  15  cwt. 
114  lb. 

244.  2/r 

245.  5  mo.  5  da. 
4  hr. 

1246.   723? 
247.   3.1 
248    404i 
264.   40,000.000  m; 

1,574.800,000  in.; 

24,854.79+  mi. 

267.  1,000,000.000,- 

000  sq  mm; 
1 .000,000  sq 
Dm 

268.  10 

269.  Lay  off  a 
square  10  m 
X  10  m 

272.    185.28  A. 

276.  1,000,000.000 
cudm;  1,000,- 
000,000  cu  Dm 

277.  Lay  off  a  cube 

1  m  X  1  m  X 
1  m 


ANSWERS 


315 


pp.  139  to  144  pp.  149  to  156  pp.  156  to  169         pp.  159  to  163 


Denominate  Num- 
bers 
pp.  i;«>-144 

280.  Wood    at   $3 
per  cd. 

281.  ^4»00 
284.  ZitAHl; 

385.(i23  cu  m 

8.192  cum 

$  180.22 

176.57+  cu.  ft. 

ex. ;    180   cu. 

ft.  approx. 
887.  73.52+bu.ex.; 

80  bu.  approx. 
838.  41.18+ Kg 

ex.;  45^,  Kg 

approx. 

1067(iO  ft. ;  20 

mi.  TOrd.ri  ft. 

3   yd.    2    ft.; 

1  rd. 

127  rd.  1  yd. 
3.6  in. ;  .3H75 
mi. 

842.  «7.r)  lb.  troy; 
72  lb.  avoir. 
l«5.8bu.;  ir)7.5 

16.9  bu. ;  157  JS 
gal. 

2  hr.  35  min. 
22  sec.;  38° 
50*  30" 

846.  25  gal.  2  at.  1 
pt. ;   6  gal.  3 

847.  Sept.  f;  Mar. 
3 

848.  Sept.  5;  Mar. 
3 

849.  283  yr.  8  mo. 
13  da. 

860.  174  da. 

861.  g000cucm;91 

862.  5000  Kg;  6cu 

868.  $13.34 
864.  962.92 


287 
288. 
886 


839. 
840. 
841. 


843. 
844. 
845. 


Literal  Quantities 

pp.  14!^15«J 

39.  3f«-3x-«+2 

40.  3a^b-bci 

51.  -i«  +  5x*y 

52.  x^y  -  arV - 

12  2T/«  +  6 

58.  2u26^  +  5ai6c 
+  6  ab^c  + 

2  fee* 

56.  «»;  x«;  ... 

57.  «7;  a*;  ... 
72.   x*  +  xiy^-\- 

y* 

75.  a6  +  5a<6  + 

10(/«62  + 

+  66 

76.  a8  +  6«-c«  + 

3  «62  +  3  a^b 
+  3ac2  — ."kiV 
— 6a6c+36c'^ 
-3  62c 

79.  a<;  a*-  ... 

80.  x;  zo,  or  1;  ... 

90.  x*-x»y+x^i 
—  xy*  +  y* ; 
5x-2y 

91.  X2  +  XV+I/2; 

x*+x^y  + 

y* 

92.  3x  +  2y;    x» 

+  zV  +  af»y« 
+  »/» 
98.  3a6(a  +  6); 
6(z«-2z2  + 
4z-l) 
94.   12a262(6-a); 
7(2x«i/-4xi/2 
+  xf/  +  2) 
96.   12«262(2a6 
-3); 
2xw(6xy— .1  V 
+4-2X) 
96.    10aS(2o&4 
-3); 
5i/(2x«-4x«v 
+2xy*-y*) 


Literal  Quantities 

pp.  15(;-1."»9 

97.  3a«62(4 

+3«62); 
6xi/(3x» 
+5x2y 
-Axy-i 

+'2yh 

103.  r)u26 

104.  2a86 

105.  a26 

106.  3«262 

107.  2'i2 

113.  12  xV 

114.  30x«v2 

115.  6x«y2 

116.  42  xV 

117.  48x«y6 
a26.    J_ 

2  '    6x 
3     .    2x 
8x6y'    3y« 
3     .    2 

5a6c'    3 

1  .  xji 
Aabc'  3 
2o62  ^ 
a262'  a262' 
462  2o? 
a-ib^'  q262 

181.    21X  +  9. 

24x 
16 X -6 

24  X 
134.   -I 

186.  9^_5y 

24 

187.  ^ 


124. 
125. 
126 
127. 


129.    ^?     '2^, 


139. 


25y 
3  a 


5  6 

140  x«8 
141.  x  =  3 
142  xr=4 
148    x  =  4 


144 


-2 


Literal  Quantities 

15«^l(i3 

145.  x  =  -2J 

146.  x  =  2 

147.  x  =  3 

148.  x  =  26 

149.  x  =  2 

150.  x  =  8 

151.  x  =  2 

152.  x  =  16 

153.  x  =  2 

155.  3()  boys 

156.  15  on  1st;  30 
on2d;*i0on3d 

157.  $(;00,  house; 
«1200,  barn; 
$  1800, land 

158.  16  horses,  48 
cows,  144 
sheep 

159.  45yr.,A'sage; 
B's.it;  C'8,27 

160.  »12,  A's 
share:  S24, 
B's;  $144, 
C's;  »84,  D's 

161.  39  in. 

162.  24,  first  part; 
8,  second ;  64, 
third 

163.  21»1 
164    $4 

167.   11,   the    less; 
i  3;^,  the  greater 

168.  rn 

169.  21  yr.,  Anna: 
2.'»  yr.,  Mary 

170.  iri 

171.  IH.  the  les-s: 
42,  the  greater 

172.  25  yr.,  son; 
45. father 

173.  3,  the  less:  31. 
the  greater 

174.  9,  the  less ;  25, 
the  greater 

176.  80,   the    less; 

100,  the 

greater 
176.  96  gal. 


816 


ANSWERS 


pp.  163  t<n68 

PP 

.  168  to  179 

pp.  179  to  184 

PP 

.184  to  189 

Literal  Quantities 

Literal  Quantities 

Proportion 

Solution  of  Prob- 

pp.  Il>3-lti8 

1(>8-I(il» 

p.  17D 

lems 
pp.  184-189 

177.    12,  first  part; 

221 

x  =  3,  y=4 

51. 

$  114,  A ; 

24. 

2  cts. 

16,  second; 

222 

3c  =  4,  y  =  3 

$:«8.  B; 

25. 

2  doz.  eggs 

13, third 

223 

x  =  2,  i/«=l 

$21U,C 

26. 

15  eggs 

178    S3 

224 

x  =  l.y  =  l 

52. 

$«r2..'i0.  A; 

27. 

84  ct«.  per  doz. 

179.   SSTiOO 

225 

X  =  6,  y  =  7 

$87.50,  B 

28. 

5  cts. 

180.   «5400 

226 

x=42.  y  =  W 

68. 

$240,  A; 

29. 

120  apples 

181.   $1500 

228 

8,12 

$320,  B; 

31. 

$2 

182.  (}yd.;9yd. 

229 

$1,  potatoes; 

$400,   C; 

32 

20  da. 

190.  x  =  6 

55  cts.,  apples 

9480,  D 

33 

$4 

191.  z  =  7 

230 

1<>    girls,     H* 

M. 

817000,  wife; 

34 

224  da. 
I2I  da. 

192.  x=8 

b(»y8 

$6000,  each 

35. 

193.  a:  =  8A 

231 

« 

son;  $4000, 

39. 

J 

194.  z=10 

232 

$50,  A;  $70, 

each  daughter 

40. 

A:  A;  21  da. 

197.  2<i4  girls 

B 

66. 

$664!,  A; 

41. 

24  da. 

198.   $5(i,  John; 

. 

$24;ii,B; 

42. 

120  da.,  A ;  60 

$42,  James 

Proportion 

$562,  C 

da.,  B;  40  da.,  C 

199.   180  acres 

pp.  173-17U 

66. 

$7128,  A; 

48. 

41  da. 

200.  itW  peach 

10. 

$18.90 

$a'i64,B; 

44. 

10  hr. 

trees 

11. 

28  eggs 

$1188,  C 

45. 

Ahr. 

201.   12yr..  Ann; 

12. 

12  men 

67. 

24  lb.  cheese 

46. 

(iOmi. 

9  yr.,  Jane 

13. 

.«  da. 

58. 

64  lb.  coffee 

47. 

90  mi. 

202.  2<>4 

14. 

lU  mo. 

48. 

$10 

203.  25  horses 

15. 

40  mi. 

Solution  of  Prob- 

49. 

$28,  A;  $35,  B 

204.   IG  ft.  wide,  22 

16. 

1*23  ft. 

lems 

50. 

2  cts.,  Henry; 

ft.  long 
205.  30  apples 

17. 

$16 

pp.  181-184 

4  cts.,  James 

18. 

1.^  bu. 

51. 

$  1680  each 

206.  $120 

19. 

■S  3750 

2. 

H  min.  space 

52. 

$6720,  A; 

207.  49,50,51 

20. 

120  men 

8. 

r^m  min.  sp. 

$3840,  B; 

208.  $140,    horse; 

21. 

2.SI  da. 

4. 

5  min.  sp. 

$10,752,0; 

$140,    car- 

22. 

Tida. 

5. 

25  sp. 

$22,400,  D 

riage 

23. 

<>  men 

6. 

5fi  min.  past  5 

57. 

5;  4 

209.   $40,    first; 

24. 

IH  hr. 

7. 

27A  min.  past  2 

58. 

15;  20 

$75,  second 

34. 

102  da. 

8. 

50;    liSmin.sp. 

%: 

40:  25 

210.   $600,  A; 

35. 

$1750 

9. 

46,^  min.  past  4 

$72 

$:ftK),  B 

36. 

750  da. 

10. 

43/i  min.  past  8 

61. 

$45 

311.  48  cts.,    first 

37. 

:muo  lb. 

11. 

12  mi. 

62 

$50 

son;  56  cts., 

38. 

28  da. 

12. 

10  hr. 

63. 

$27 

second;    39 

39. 

6  da. 

13. 

60  mi. 

64. 

$8 

cts.,  third 

40. 

55.125  lb. 

14. 

.%  mi. 

65. 

$60.  cost; 

212.   $325 

41. 

7hr. 

15. 

12  mi..   A;    14 

$  100,  selling  p. 

213.  50  A,  A ;  65,  B: 

42. 

4k  ft. 

mi.,B 

66. 

$200,  cost; 

115,  C ;  33,  D 

43. 

$24 

17. 

1    hound    leap 

$  180,  selling  p. 

215.   x  =  l.  y  =  2 

44 

.>40  pp. 

=  5  fox  leaps 

67. 

J 

216.  2  =  2,  y  =  3 

45 

24  Iwys 

18 

1  fox  leap 

68. 

1 

217.  x=-l,  v  =  4 

49. 

.■*  102, 1st  part; 

19. 

72  times 

69. 

$540 

218.   x  =  6,  y  =2 

$ln3,  2d  part: 

20. 

252  times 

70. 

30  ct. 

219.   «=-!,  v  =  2 

$2.5.5,  3d  part 

21. 

m  leaps 

71. 

$10,  com.; 

220.  x  =  12,  y  =  4 

50. 

$1,t5 

23. 

let. 

$190,  proceeds 

ANSWERS 


817 


318 


ANSWERS 


pp.  218  to  227 

PP 

.  227  to  235 

PP 

.  235  to  245 

pp.  247  to  256 

Interest 

Interest 

Interest 

Involution  and 

pp.  218-227 

1 

pp.  227-235 

pp.  235-2:>7 

Evolution 

24.  $84.61 

85. 

$.300 

133. 

$733.99 

pp.  247-248 

25.   S1.48 

86. 

$2a'5.97 

134. 

$  1748.28 

60. 

460 

26.  S6.51 

87. 

$114.19 

138. 

$271.2.3 

61. 

562 

27.   «5.48 

88. 

$  1(».-..8G 

139. 

$261.19 

62. 

1022 

28.  $6.16 

89. 

$i;J7.18 

63. 

1013 

29.  $5.26 

93. 

Due  March  10; 

67. 

.1539+ 

34.   $4.57 

$4,  bank  dis. 

Involution  and 

68. 

.8735+ 

35.   $42.43 

94. 

$1.60 

Evolution 

69. 

i 

36.   $02.88 

95. 

$1.:J4 

pp.  239-245 

70. 

.12 

87.   $17.26 

96. 

$3.96,    true 

71. 

.a5569+ 

48.   10% 

dis. ;    banker 

14. 

32 

72. 

.<K).J7+ 

49.  2  yr.  3  mo.  18 

makes  4  ^ 

16. 

45 

73. 

1.2 

ila. 

97. 

Int.  on  $3.96 

17. 

24 

74. 

5.rS9+ 

50.  20  yr. 

(K)da.  is  4^ 

19. 

95 

75. 

.7r.39+ 

51.  2yr.  3  mo.  18 

98. 

$227.25. 

20. 

97 

76. 

.02585+ 

da. 

99. 

$744.08,    face 

21. 

83 

77. 

.025 

52.  5% 

of  note 

22. 

37 

79. 

a6+(5a*+ 

53.  5% 

104. 

$«;i6.18 

23. 

48 

10a«6+10a26a 

54.  $260 

105. 

$<!l(i.l8 

24. 

85 

+5o&«+6<)6 

55.  $2(K) 

106. 

June  9 

25. 

91 

80. 

a«+(6a6+ 

56.  $24.60 

107. 

.lune  20 

26 

32 

15  0*6+20  a862 

57.  $3332 

108. 

^.•W.25,int.on 

27. 

73 

+  15a26»+6a64 

58.   $1313.78 

nmt.  of  debts: 

29. 

314 

+6«)& 

59.  $158.60 

$20.10.  int.  on 

30. 

231 

81. 

12J 

60.   $660 

ami.  of   pav- 

31. 

625 

82. 

24 

61.   $6000 

m'ts;    $16.1*6, 

32. 

723 

83. 

311 

62.  $500 

dif. 

33. 

1039 

84. 

12 

68.   $980 

110. 

$85.51 

84. 

1829 

64.  5% 

111. 

$34.16 

38. 

.060415+ 

65.  6% 

112. 

$59.73 

39. 

.8KH9+ 

Mensuration 

66.  3J% 

lis. 

$129.86 

40. 

! 

p.  256 

67.  2% 

114. 

$1874.11 

41. 

.25 

68.  3  yr.  6  mo. 

115. 

$3262.50 

42. 

.790.'i6+ 

57. 

50.2656  in. 

69.  5yr. 

116. 

$918.72 

43. 

.4714+ 

58. 

37.<;992  in. 

70.  9  mo.  18  da. 

117. 

$135.5.16 

44. 

2.5 

59. 

62.832  ft 

71,  3  yr.  9  mo. 

119. 

$(1.06)S 

45. 

7.9056+ 

60. 

6ift. 

72.  $175 

$1.262477 

46. 

.65465+ 

61. 

80  in. 

73.   $130 

122. 

$(1.06)2«)o. 

47. 

.0079056+ 

62. 

2im 

74.   $256 

4000   decimal 

48. 

.025 

63. 

10  in. 

76.  $933.33 

places 

50. 

23 

64. 

2(jin. 

77.  $143.05 

125. 

82999.15 

51. 

34 

65. 

40  in. 

78.   $2<).73 

126. 

$88.27 

52. 

49 

66. 

20  in. 

80.   $808.11 

127 

$278.52 

63. 

74 

67. 

24111. 

81.   $72.6<> 

128. 

$381.18 

64. 

68 

68. 

5  in. 

83.   §2'.)7.44   pres. 
worth;   $2.56 

129. 

$19.94 

66. 

63 

69. 

24  in. 

130. 

.5  18.57 

66. 

89 

70. 

28  in. 

true  dis. 

131. 

$9.45 

67. 

53 

71. 

40  ft. 

84.   $400 

132. 

$.5^0.12 

68. 

55 

72. 

119  ft. 

ANS\ 

VEI 

IS 

319 

pp.  866  to  867 

PP 

.  867  to  281 

pp.  881  to  884 

PP 

.  884  to  888 

Mensuration 

Mensuration 

Occupations 

Occupations 

pp.  •J5(>--JG7 

pp.  2(J7-271 

pp.  281-284 

pp.  284-288 

78.  374  ft. 

167. 

WAim  sq  m 

34. 

3<K)bd.  ft. 

74. 

24  ft.of  boards 

74.  6  ft.  3  ill. 

168. 

10.31+  m 

35. 

lAO  IhI.  ft. 

76. 

120    ft.    of 

75.   13.200  ft. 

169. 

12  n» 

36. 

48  1x1.  ft. 

boards 

76.   ;i4ft. 

170. 

rA).'2tM  sq.  in. 

37. 

90  bd.  ft.    See 

76. 

341    ft.    of 

77.   18  ft. 

171. 

13,824  sq.  in. 

Note. 

boards 

88.   l.n.08sq.  m. 

182. 

16!  1.6461111.  in. 

38. 

108  M.  ft. 

78. 

1  sq.  yd. 

84.   lOOni. 

183. 

3<Jin. 

39. 

144  bd.  ft. 

79. 

Crosswise ; 

85.  4A.1108q.rd. 

184. 

40  sq.  in. 

40. 

144  bd.  ft. 

makes  no  dif- 

86.  200  so.  111. 

185. 

4  times 

41. 

240  bd.  ft. ;  480 

ference  where 

87.  :«..Y.+  ft. 

186. 

16 

bd.  ft. 

the  figure  be- 

88.   13A.208q.rd. 

187. 

4  lb. 

48. 

16  ft. ;  12  ft. 

gins 
1 33.75 

89.  *.^>  sq.  dm 

188. 

1:4 

43. 

24(»  bd.  ft. 

80. 

90.   1  rd. 

189. 

Sin. 

44. 

22r)  b<i.  ft. 

81. 

1     yd.    wide, 

91.  27  8qm 

190. 

S4. 

45. 

2240  ft. 

lengthwise, 

98.  4i  A. 

191. 

4000  11). 

46. 

S24.(r) 

$26! 

98.   3I4.ir.sqm 

192. 

4S0  bu. 

47. 

1500  bd.  ft. 

88. 

S  39.06 

94.  :m.w2  sq.  ft. 

193. 

(J400CU.  ft. 

48. 

Agrees 

83. 

367.no 

95.   Imi. 

194. 

4  in. 

49. 

•Ml  bd.  ft. 

85. 

6i    double 

96.   70.246+ rd. 

195. 

8<'4     ct..     A; 

50. 

Agrees 

rolls 

97.    173.2    sq.   ft., 

121ct.,B 

51. 

IK)  sq.  in. 

86 

2  strips 

area;  17.32  ft. 

52. 

3ii  .sq.  yd. 

87. 

3  strips 

98.   !NiO  sq.  ft. ;  48 
ft. 

Occupations 

53. 
54. 

Agrees  nearlv 
3«)  bundles,  by 

88. 

5i    d(mble 
rolls 

99.    10,000  sq.  ft. 

pp.  280-281 

rule 

89. 

41   double 

100.    113.0ir7(>      sq. 

55. 

27J  sq.  ft. 

rolls 

ft.;  8:10 

1. 

8bd.  ft. 

56. 

Agrees  nearly 

90. 

$6.48    (strips 

148.   48  sq.  in. 

8. 

4  bd.  ft. 

57. 

31i  sq.  ft. 

on  ceiling  run 

144.   62.8;«  sq.  ft. 

4 

9  bd.  ft. 

68. 

Agrees  nearly 
61  bunches,  by 

lengthwise) 

145.   l.'i0.71HHi  sq.  in. 

5. 

12  bd.  ft. 

59. 

91. 

No ;    strips 

146.  452.3904 sq.ft. 

147.  314.16  sq.  in. 

6. 

15  b«l.  ft. 

rule  ;    rafters, 

must     match 

8. 

H  bd.  ft. 

17.77  ft.  Ion- 

and  be  whole 

148.  288     sq.     in.. 

10. 

12  ImI.  ft. 

60. 

.•«»20ft. 

06. 

1(H.18  bbl.; 

entire  surface 

11. 

20  bd.  ft. 

61. 

320  bd.  ft. 

57.87  cu.  yd. 

149.  4  in. 

12. 

27  bd.  ft. 

62. 

[m  bd.  ft. 

07. 

5  masons 

150.  50.26S6in. 

14. 

:w4  bd.  ft. 

63. 

12  sq.   yd.,  bv 

08. 

1.^.7,520  bricks 

151.   12  ft. 

15. 

40bd.ft. 

rule  ;    VM   sq 

99. 

S  748.44  ; 

158.  8.6<!+ft. 

16. 

21  b<l.  ft. 

yd.,  by  com. 

charge  for tho 

168.  576  sqm 

17. 

42  bd.  ft. 

64. 

100  sq.  ft.,  by 

openings 

164     673.87  sq.  ft. 
155.  201,0lK>,fo0sq. 

18. 

240  b<l.  ft. 

rule;   llli  sq. 

100. 

W.45  cu.  yd.; 

19. 

»;o  bd.  ft. 

ft.                    * 

177.21  bbl.; 

mi. 

20. 

UK)bd.  ft. 

65. 

.32  bd.  ft. 

1.^4  da. 

166.  280 sq.ft. 

27 

;!jbd.  ft. 

67. 

120  1x1.  ft. 

101. 

8800  bricks 

167.   S7.fi0 

28 

i(;4)b(i.  ft. 

68. 

100  M.  ft. 

108. 

2M7  bricks 

168.  «R».28cu.  in. 

29. 

72  M.  ft. 

69. 

432  bd.  ft. 

103. 

9:t87  bricks 

168.   ll.m»76(u.in. 

SO. 

192  1x1.  ft. 

70. 

96  ft.  of  l>oards 

104. 

The    number 

164.  62.362  cu.  in. 

31. 

11>2  ImI.  ft. 

78. 

60  ft.  of  boartls 

of   cu.  ft.  in 

166.  565.^88  cu.  in. 

88. 

•Mi  M.  ft. 

78. 

115    ft.    of 

walls      is     4 

166.  20  iu. 

83. 

VA  bd.  ft. 

boards 

times  as  many 

320 

ANSWERS 

pp.  288  to  289          pp 

.  289  to  291         pp.  291  to  298         pp.  293  to  294 

Occupations 

Occupations 

Occupations 

Miscellaneous 

pp.  288-289 

pp.  289-291 

p.  291 

pp.  293-2iM 

105.  42  cd. 

117. 

85+ bu. 

181.  322  rd. 

4. 

6   hr.  45   min. 

106.    ItiiiJ?  i>erch 

118. 

12Hbu.             !1S2.  50.5(i+  rd. 

•M  sec. 

107.   r»2i    bbl.;     42 

119. 

4+  bins 

133.  44.8  rd. 

78^  27' 35"   W. 

cu.  yd. 

120. 

2bu. 

184.  80 trees;  each 

* 

3ftfg  sec.  past 

108.  Si« 

122. 

2  ft.  1  in. 

tree  the  cen- 

2.30 A.M. 

109.    Ill  bbl.;  4 cu. 

123. 

282.7+ bbl.; 

ter  of  a  2  rd. 

Ihr. 

.vd. 

count  7i  gal. 

square 

West 

110.  4fiO|bu. 

1  cu.  ft. 

136.   15  ft.  long 

100  days 

111.  36841  bu. 

125. 

24  T. 

186.  40 bbl.;  126 hr. 

7.7 

112.   460tbu.; 

126. 

6Ami. 

48U  lb. 

25,804.8  lb. 

127. 

7Arai. 

i 

113.  5r»5?    bu.    ap- 

128. 

3240  bills; 

37nb. 

proximately 

each  hill  the 

2i 

114.  7<i+bu. 

center  of  a  3 

Miscellaneous 

20^C.;16°R. 

115.   fK*)]?    bu.    ap- 

ft. 8  in.  square 

p.  293 

635°  F. 

proximately 

129. 

21i  bo.  per  A. 

1000°  C. 

116.  :«)bu.  old;28  180. 

Agrees  nearly 

8.  101°  22' 30" 

18. 

172rF.;62rR. 

bu.  new 

20^  R. 

re  35799 


